Compressed lattice sums arising from the Poisson equation

  • David H Bailey1 and

    Affiliated with

    • Jonathan M Borwein2Email author

      Affiliated with

      Boundary Value Problems20132013:75

      DOI: 10.1186/1687-2770-2013-75

      Received: 4 December 2012

      Accepted: 15 March 2013

      Published: 5 April 2013

      Abstract

      Purpose

      In recent years attention has been directed to the problem of solving the Poisson equation, either in engineering scenarios (computational) or in regard to crystal structure (theoretical).

      Methods

      In (Bailey et al. in J. Phys. A, Math. Theor. 46:115201, 2013, doi:10.1088/1751-8113/46/11/115201) we studied a class of lattice sums that amount to solutions of Poisson’s equation, utilizing some striking connections between these sums and Jacobi ϑ-function values, together with high-precision numerical computations and the PSLQ algorithm to find certain polynomials associated with these sums. We take a similar approach in this study.

      Results

      We were able to develop new closed forms for certain solutions and to extend such analysis to related lattice sums. We also alluded to results for the compressed sum

      ϕ 2 ( x , y , d ) : = 1 π 2 m , n O cos ( π m x ) cos ( π n d y ) m 2 + d n 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ1_HTML.gif
      (1)

      where d > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq1_HTML.gif, x, y are real numbers and O http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq2_HTML.gif denotes the odd integers. In this paper we first survey the earlier work and then discuss the sum (1) more completely.

      Conclusions

      As in the previous study, we find some surprisingly simple closed-form evaluations of these sums. In particular, we find that in some cases these sums are given by 1 / π log A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq3_HTML.gif, where A is an algebraic number. These evaluations suggest that a deep theory interconnects all such summations.

      PACS Codes:02.30.Lt, 02.30.Mv, 02.30.Nw, 41.20.Cv.

      MSC:06B99, 35J05, 11Y40.

      Keywords

      lattice sums Poisson equation experimental mathematics high-precision computation

      1 Background

      In [1], we analyzed various generalized lattice sums [2], which have been studied for many years in the mathematical physics community, for example, in [24]. More recently interest was triggered by some intriguing research in image processing techniques [5]. These developments have underscored the need to better understand the underlying theory behind both lattice sums and the associated Poisson potential functions.

      In recent times, attention was directed to the problem of solving the Poisson equation, either in engineering scenarios (computationally, say for image enhancement) or in regard to crystal structure (theoretically). In [1] we thus studied a class of lattice sums that amount to Poisson solutions, namely the n-dimensional forms
      ϕ n ( r 1 , , r n ) = 1 π 2 m 1 , , m n odd e i π ( m 1 r 1 + + m n r n ) m 1 2 + + m n 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equa_HTML.gif
      By virtue of striking connections with Jacobi ϑ-function values, we were able to develop new closed forms for certain values of the coordinates r k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq4_HTML.gif and extend such analysis to similar lattice sums. A primary result was that for rational x, y, the natural potential ϕ 2 ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq5_HTML.gif is 1 π log A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq6_HTML.gif where A is an algebraic number. Various extensions and explicit evaluations were given. We also touched on results for the compressed sum
      ϕ 2 ( x , y , d ) : = 1 π 2 m , n O 2 cos ( π m x ) cos ( π n d y ) m 2 + d n 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ2_HTML.gif
      (2)

      In this paper we discuss (2) in more detail. More detailed motivation can be found in [1, 5].

      Such work is made possible by number-theoretical analysis, symbolic computation and experimental mathematics, including extensive numerical computations using up to 25,000-digit arithmetic.

      In Section 1.1, we describe, consistent with [5], the underlying equations along with ‘natural’ Madelung constants and relate them to the classical Madelung constants. In Section 1.2, we produce the solutions ϕ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq7_HTML.gif which, especially with n = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq8_HTML.gif, provide the central objects of our study. In Section 2.1, we describe rapid methods of evaluating ϕ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq9_HTML.gif and add computational details in Section 2.2. In Section 3.1 we discuss closed forms for ϕ 2 ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq5_HTML.gif. In Section 3.2 we discuss closed forms for related sums in two dimensions. This leads to the introduction of theta function formalism in the next two subsections and allows us to resolve much else. In Sections 3.5 and 3.6 we are then able to evaluate compressed potential sums. Finally, we further discuss computational matters in Section 3.7.

      1.1 Madelung-type sums

      In a recent treatment of ‘natural’ Madelung constants [5], it is pointed out that the Poisson equation for an n-dimensional point-charge source,
      2 Φ n ( r ) = δ ( r ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ3_HTML.gif
      (3)
      gives rise to an electrostatic potential - we call it the bare-charge potential - of the form
      Φ n ( r ) = Γ ( n / 2 1 ) 4 π n / 2 1 r n 2 = : C n r n 2 , if  n 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ4_HTML.gif
      (4)
      Φ 2 ( r ) = 1 2 π log r = : C 2 log r , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ5_HTML.gif
      (5)
      where r : = | r | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq10_HTML.gif. Since this Poisson solution generally behaves as r 2 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq11_HTML.gif, the previous work [5] defines a ‘natural’ Madelung constant N n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq12_HTML.gif as (here, m : = | m | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq13_HTML.gif)
      N n : = C n m Z n ( 1 ) 1 m m n 2 , if  n 2 , N 2 : = C 2 m Z n ( 1 ) 1 m log m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ6_HTML.gif
      (6)
      where in cases such as this log sum one must infer an analytic continuation [5], as the literal sum is quite non-convergent. This N n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq12_HTML.gif coincides with the classical Madelung constant
      M n : = m Z n ( 1 ) 1 m m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ7_HTML.gif
      (7)

      only for n = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq14_HTML.gif dimensions, in which case N 3 = 1 4 π M 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq15_HTML.gif. But in all other dimensions there is no obvious ℳ, N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq16_HTML.gif relation.

      A method for gleaning information about N n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq12_HTML.gif is to contemplate the Poisson equation with a crystal charge source, modeled after NaCl (salt) in the sense of alternating lattice charges:
      2 ϕ n ( r ) = m Z n ( 1 ) 1 m δ ( m r ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ8_HTML.gif
      (8)
      Accordingly - based on the Poisson equation (3) - solutions ϕ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq7_HTML.gif can be written in terms of the respective bare-charge functions Φ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq17_HTML.gif as
      ϕ n ( r ) = m Z n ( 1 ) 1 m Φ n ( r m ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ9_HTML.gif
      (9)

      1.2 The crystal solutions ϕ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq7_HTML.gif

      In [5] it is argued that a solution to (8) is
      ϕ n ( r ) = 1 π 2 m O n k = 1 n cos ( π m k r k ) m 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ10_HTML.gif
      (10)

      where O http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq18_HTML.gif denotes the odd integers (including negative odds). These ϕ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq7_HTML.gif do give the potential within the appropriate n-dimensional crystal, in that ϕ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq7_HTML.gif vanishes on the surface of the cube [ 1 / 2 , 1 / 2 ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq19_HTML.gif, as is required via symmetry within an NaCl-type crystal of any dimension - thus we have solved the Poisson equation subject to Dirichlet boundary conditions.

      To render this representation more explicit and efficient, we could write equivalently
      ϕ n ( r ) = 2 n π 2 m 1 , , m n > 0 , odd cos ( π m 1 r 1 ) cos ( π m n r n ) m 1 2 + + m n 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equb_HTML.gif
      It is also useful that - due to the symmetry inherent in having odd summation indices - we can cavalierly replace the cosine product in (10) with a simple exponential:
      ϕ n ( r ) = 1 π 2 m O n e i π m r m 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ11_HTML.gif
      (11)

      This follows from the simple observation that exp ( π i m k r k ) = ( cos ( π m k r k ) + i sin ( π m k r k ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq20_HTML.gif, so when the latter product is expanded out, the appearance of even a single sin term is annihilating due to the bipolarity of every index m k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq21_HTML.gif.

      We observe that the convergence of these conditionally convergent sums is by no means obvious, but that results such as [[2], Thm. 8.3 and Thm. 8.5] ensure that
      ϕ n ( r , s ) : = 1 π 2 m O n k = 1 n cos ( π m k r k ) m 2 s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ12_HTML.gif
      (12)

      is convergent and analytic with abscissa σ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq22_HTML.gif for ( n 1 ) / 4 σ 0 ( n 1 ) / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq23_HTML.gif, where Re ( s ) = σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq24_HTML.gif. For the central case herein, summing over increasing spheres is analytic in two dimensions for σ 0 23 / 73 < 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq25_HTML.gif and in three dimensions for σ 0 25 / 34 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq26_HTML.gif, but in general the best estimate we have is σ 0 n / 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq27_HTML.gif, so for n 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq28_HTML.gif, to avoid ambiguity, we work with the analytic continuation of (12) from the region of absolute convergence with σ > n / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq29_HTML.gif. Indeed, all our transform methods are effectively doing just that.

      2 Methods

      2.1 Fast series for ϕ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq7_HTML.gif

      From previous work [5] we know a computational series
      ϕ n ( r ) = 1 2 π R O n 1 sinh ( π R ( 1 / 2 | r 1 | ) ) k = 1 n 1 cos ( π R k r k + 1 ) R cosh ( π R / 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ13_HTML.gif
      (13)
      suitable for any nonzero vector r [ 1 / 2 , 1 / 2 ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq30_HTML.gif. The previous work also gives an improvement in the case of n = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq31_HTML.gif dimensions, namely the following form for which the logarithmic singularity at the origin has been siphoned off:
      ϕ 2 ( x , y ) = 1 4 π log cosh ( π x ) + cos ( π y ) cosh ( π x ) cos ( π y ) 2 π m O + cosh ( π m x ) cos ( π m y ) m ( 1 + e π m ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ14_HTML.gif
      (14)

      These series, (13) and (14) are valid, respectively, for r 1 , x [ 1 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq32_HTML.gif.

      For clarification, we give here the ( n = 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq33_HTML.gif-dimensional case of the fast series:
      ϕ 3 ( x , y , z ) = 2 π p , q > 0 , odd sinh ( π 2 p 2 + q 2 ( 1 2 | x | ) ) cos ( π p y ) cos ( π q z ) p 2 + q 2 cosh ( π 2 p 2 + q 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ15_HTML.gif
      (15)

      Though it may not be manifest in this asymmetrical-looking series, it turns out that for any dimension n the ϕ n ( r 1 , , r n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq34_HTML.gif is invariant under permutations and sign-flips. For example, ϕ 3 ( x , y , z ) = ϕ 3 ( y , z , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq35_HTML.gif and so on. It thus behoves the implementor to consider x - which appears only in the first sum of (15) - to be the largest in magnitude of the three coordinates for optimal convergence. A good numerical test case is the exact evaluation ϕ 3 ( 1 / 6 , 1 / 6 , 1 / 6 ) = 3 4 π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq36_HTML.gif, which we have confirmed to 500 digits.

      2.2 Computational techniques

      In this study, we employ an experimental scheme similar to that used in [1], as well as in numerous other studies that have been performed. In particular, we compute the key expressions in this study to very high precision, then use the PSLQ algorithm in an attempt to recognize the computed numerical values in terms of relatively simple, closed-form expressions. Given an n-long input vector ( x i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq37_HTML.gif) of real numbers, the PSLQ algorithm finds a nontrivial set of integer coefficients ( a i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq38_HTML.gif) such that a 1 x 1 + a 2 x 2 + + a n x n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq39_HTML.gif if it exists (to within the tolerance of the numeric system being used).

      Numerous experimental evaluations of minimal polynomials satisfied by exp ( 8 π ϕ 2 ( x , y ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq40_HTML.gif (the case d = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq41_HTML.gif), and a detailed description of the underlying computational methodology employed to find these evaluations, are presented in [1]. A similar computational scheme was employed in this study to obtain minimal polynomials satisfied by exp ( 8 π d ϕ 2 ( x , d y , d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq42_HTML.gif. For instance, we recover, among a myriad of other evaluations,
      ϕ 2 ( 1 3 , 1 3 , 9 ) = 1 24 π log ( 3 + 2 3 9 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equc_HTML.gif
      This computational scheme is as follows:
      1. 1.

        Given x, y and d, select a conjectured polynomial degree m and a precision level P. We typically set the numeric precision level P somewhat greater than m 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq43_HTML.gif digits.

         
      2. 2.

        Compute ϕ 2 ( x , y d , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq44_HTML.gif to P-digit precision using formula (47). Evaluate the four theta functions indicated using the very rapidly convergent formulas given in [[6], p.52] or [7].

         
      3. 3.

        Generate the ( m + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq45_HTML.gif-long vector ( 1 , α , α 2 , , α m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq46_HTML.gif, where α = exp ( 8 π d ϕ 2 ( x , y d , d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq47_HTML.gif. Note: we have found that without the eight here, the degree of the resulting polynomial would be eight times as high (but the larger polynomials were in fourth or eighth powers). Given the very rapidly escalating computational cost of higher degrees, many of the results in [1] and herein would not be feasible without this factor.

         
      4. 4.

        Apply the PSLQ algorithm (we employed the two-level multipair variant of PSLQ for d up to 19, and the three-level multipair variant for d 20 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq48_HTML.gif [8]) to find a nontrivial ( m + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq45_HTML.gif-long integer vector A = ( a 0 , a 1 , a 2 , , a m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq49_HTML.gif such that a 0 + a 1 α + a 2 α 2 + + a m α m = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq50_HTML.gif, if such a vector exists. PSLQ (or one of its variants) either finds a vector A, which then is the vector of coefficients of an integer polynomial satisfied by α (certified to the ‘epsilon’ of the numerical precision used), or else exhausts precision without finding a relation, in which case the algorithm nonetheless provides a lower bound on the Euclidean norm of the coefficients of any possible degree-m integer polynomial A satisfied by α.

         
      5. 5.

        If no numerically significant relation is found, try again with a larger value of m and correspondingly higher precision. If a relation is found, try with somewhat lower m, until the minimal m is found that produces a numerically significant relation vector A. Here ‘numerically significant’ means that the relation holds to at least 100 digits beyond the level of precision required to discover it. To obtain greater assurance that the polynomial produced by this process is in fact the minimal polynomial for α, use the polynomial factorization facilities in Mathematica or Maple to attempt to factor the resulting polynomial.

         

      3 Results and discussion

      3.1 Closed forms for the ϕ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq51_HTML.gif potential

      Provably we have the following results which were established by factorization of lattice sums after being empirically discovered by the methods described in the next few sections.

      Theorem 1 ([1])

      We have
      ϕ 2 ( 1 / 3 , 1 / 3 ) = 1 8 π log ( 1 + 2 / 3 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ16_HTML.gif
      (16)
      ϕ 2 ( 1 / 4 , 1 / 4 ) = 1 4 π log ( 1 + 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ17_HTML.gif
      (17)
      ϕ 2 ( 1 / 3 , 0 ) = 1 8 π log ( 3 + 2 3 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ18_HTML.gif
      (18)
      Using the integer relation method PSLQ [9] to hunt for results of the form
      exp ( π ϕ 2 ( x , y ) ) = ? α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ19_HTML.gif
      (19)

      for α algebraic, we may obtain and further simplify many results as follows.

      Conjecture 2 ([1])

      We have discovered and subsequently proven
      ϕ 2 ( 1 / 4 , 0 ) = ? 1 4 π log α , where α + 1 / α 2 = 2 + 1 , ϕ 2 ( 1 / 5 , 1 / 5 ) = ? 1 8 π log ( 3 + 2 5 + 2 5 + 2 5 ) , ϕ 2 ( 1 / 6 , 1 / 6 ) = ? 1 4 π log γ , where γ + 1 / γ 2 = 3 + 1 , ϕ 2 ( 1 / 3 , 1 / 6 ) = ? 1 4 π log τ , where τ 1 / τ 2 = ( 2 3 3 ) 1 / 4 , ϕ 2 ( 1 / 8 , 1 / 8 ) = ? 1 4 π log ( 1 + 2 2 2 4 1 ) , ϕ 2 ( 1 / 10 , 1 / 10 ) = ? 1 4 π log μ , where μ + 1 / μ 2 = 2 + 5 + 5 + 2 5 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equd_HTML.gif

      where the notation = ? http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq52_HTML.gif indicates that we originally only had experimental (extreme-precision numerical) evidence of an equality.

      Such hunts are made entirely practicable by (14). Note that for general x and y we have ϕ 2 ( y , x ) = ϕ 2 ( x , y ) = ϕ 2 ( x , 1 y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq53_HTML.gif, so we can restrict searches to 1 / 2 > x y > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq54_HTML.gif.

      Remark 3 (Algebraicity)

      In light of our current evidence, we conjectured that for x, y rational,
      ϕ 2 ( x , y ) = ? log α π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ20_HTML.gif
      (20)

      for α algebraic. Theorem 5 proved this conjecture. We note that Theorem 5 proves that all values should be algebraic but does not, a priori, establish the precise values we have found. This will be addressed in Section 3.4.

      3.2 Madelung and ‘jellium’ crystals, and Jacobi ϑ-functions

      We have studied
      ϕ 2 ( x , y ) : = 1 π 2 a , b O cos ( π a x ) cos ( π b y ) a 2 + b 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ21_HTML.gif
      (21)
      as a ‘natural’ potential for n = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq8_HTML.gif dimensions in the Madelung problem. There is another interesting series, namely
      ψ 2 ( x , y ) : = 1 4 π 2 a , b Z cos ( 2 π a x ) cos ( 2 π b y ) a 2 + b 2 = 1 π 2 a , b E cos ( π a x ) cos ( π b y ) a 2 + b 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ22_HTML.gif
      (22)

      where E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq55_HTML.gif denotes the even integers.

      Now it is explained and illustrated in [5] that this ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq56_HTML.gif function is the ‘natural’ potential for a classical jellium crystal and relates to Wigner sums [2]. This involves a positive charge at every integer lattice point, in a bath - a jelly - of uniform negative charge density. As such, the ψ functions satisfy a Poisson equation but with different source term [5]. Note, importantly, that ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq56_HTML.gif satisfies Neumann boundary conditions on the faces of the Delord cube - in contrast with the Dirichlet boundary conditions satisfied in the Madelung case.

      Briefly, a fast series for ψ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq57_HTML.gif has been worked out [5] as follows:
      ψ n ( r ) = 1 12 + 1 2 r 1 2 1 2 | r 1 | + 2 n 3 π S Z + ( n 1 ) cosh ( π S ( 1 2 | r 1 | ) ) k = 1 n 1 cos ( 2 π S k r k + 1 ) S sinh ( π S ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ23_HTML.gif
      (23)

      valid on the Delord n-cube, i.e., for r [ 1 / 2 , 1 / 2 ] n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq58_HTML.gif.

      In the following, we shall also use the general form of the general Jacobi theta functions, defined as in [10] and [[6], Sec. 2.6]:
      ϑ 1 ( z , q ) = 2 n = 0 ( 1 ) n q ( n + 1 / 2 ) 2 sin ( ( 2 n + 1 ) z ) , ϑ 2 ( z , q ) = 2 n = 0 q ( n + 1 / 2 ) 2 cos ( ( 2 n + 1 ) z ) , ϑ 3 ( z , q ) = 1 + 2 n = 0 q n 2 cos ( 2 n z ) , ϑ 4 ( z , q ) = 1 + 2 n = 0 ( 1 ) n q n 2 cos ( 2 n z ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ24_HTML.gif
      (24)
      One useful relation is
      ϑ 4 2 ( z , q ) ϑ 4 2 ( 0 , q ) = ϑ 3 2 ( z , q ) ϑ 3 2 ( 0 , q ) ϑ 2 2 ( z , q ) ϑ 2 2 ( 0 , q ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ25_HTML.gif
      (25)

      3.3 Closed forms for ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq56_HTML.gif and ϕ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq51_HTML.gif

      Using the series (23) for high-precision numerics, it was discovered (see [[5], Appendix]) that previous lattice-sum literature had concealed a longtime typographical error for certain two-dimensional sums, and that a valid closed form for ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq59_HTML.gif is actually
      ψ 2 ( x , y ) = x 2 2 + 1 4 π log ( Γ ( 1 / 4 ) 8 π Γ ( 3 / 4 ) ) 1 2 π log | ϑ 1 ( π ( i x + y ) , e π ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ26_HTML.gif
      (26)
      In the Madelung case, it then became possible to cast ϕ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq51_HTML.gif likewise in closed form, namely
      ϕ 2 ( x , y ) = 1 4 π log | α ( z ) | , where  α ( z ) : = ϑ 2 2 ( z , q ) ϑ 4 2 ( z , q ) ϑ 1 2 ( z , q ) ϑ 3 2 ( z , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ27_HTML.gif
      (27)

      for q : = e π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq60_HTML.gif, z : = π 2 ( y + i x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq61_HTML.gif. (See [5] and [[1], Appendix II] for details.)

      Theorem 4 ([1])

      For z : = π 2 ( y + i x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq62_HTML.gif,
      ϕ 2 ( x , y ) = 1 2 π log | ϑ 2 ( z , q ) ϑ 4 ( z , q ) ϑ 1 ( z , q ) ϑ 3 ( z , q ) | = 1 4 π log | 1 λ ( z ) 2 1 1 / λ ( z ) 2 | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ28_HTML.gif
      (28)
      and
      ψ 2 ( x , y ) = 1 4 π log | 2 μ ( 2 z ) ( 2 λ ( 2 z ) 1 ) | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ29_HTML.gif
      (29)
      where
      λ ( z ) : = ϑ 4 2 ( z , e π ) ϑ 3 2 ( z , e π ) = n = 1 ( 1 2 cos ( 2 z ) q 2 n 1 + q 4 n 2 ) 2 ( 1 + 2 cos ( 2 z ) q 2 n 1 + q 4 n 2 ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ30_HTML.gif
      (30)
      and
      μ ( z ) : = e π x 2 / 2 ϑ 3 2 ( z , e π ) ϑ 3 2 ( 0 , e π ) = q x 2 / 2 n = 1 ( 1 + 2 cos ( 2 z ) q 2 n 1 + q 4 n 2 ) 2 ( 1 + q 2 n 1 ) 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ31_HTML.gif
      (31)

      with q : = e π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq60_HTML.gif.

      We recall the general ϑ-transform giving for all z with Re t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq63_HTML.gif
      ϑ 3 k ( π z , e t π ) = 1 / t e π z 2 / t ϑ 3 + k ( i π z / t , e π / t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ32_HTML.gif
      (32)
      for k = 1 , 0 , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq64_HTML.gif (while ϑ 1 ( π z , e t π ) = 1 / t e π z 2 / t ϑ 1 ( i π z / t , e π / t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq65_HTML.gif). In particular, with t = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq66_HTML.gif we derive that
      ϑ 3 k ( π ( i x + y ) , e π ) = e π z 2 ϑ 3 + k ( π ( i y x ) , e π ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ33_HTML.gif
      (33)
      which directly relates | μ ( π ( y + i x ) ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq67_HTML.gif and | μ ( π ( x + i y ) ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq68_HTML.gif in (31). Let
      κ ( z ) : = ϑ 2 2 ( z , e π ) ϑ 3 2 ( z , e π ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ34_HTML.gif
      (34)
      Then
      κ ( i x + y ) + λ ( i x + y ) = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ35_HTML.gif
      (35)
      or 1 2 κ ( i x + y ) = 2 λ ( i x + y ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq69_HTML.gif, and (32) then shows
      λ ( i x + y ) = κ ( x + i y ) = 2 λ ( x + i y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ36_HTML.gif
      (36)

      Likewise (28) is unchanged on replacing λ by κ. Hence, it is equivalent to (20) to prove that for all z = π 2 ( y + i x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq70_HTML.gif with x, y rational λ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq71_HTML.gif in (30) is algebraic.

      Theorem 5 (Algebraic values of λ and μ, [1])

      For all z = π 2 ( y + i x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq70_HTML.gif with x, y rational, the values of λ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq71_HTML.gif and μ ( 2 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq72_HTML.gif in (30) are algebraic. It follows that ϕ 2 ( x , y ) = 1 4 π log α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq73_HTML.gif with α algebraic. Similarly, ψ 2 ( x , y ) = 1 4 π log β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq74_HTML.gif for β algebraic.

      This type of analysis also works for any singular value τ = d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq75_HTML.gif and q = exp ( 2 π i τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq76_HTML.gif and so applies to sums with n 2 + d m 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq77_HTML.gif in the denominator, as we see in the next section.

      3.4 Explicit equations for degree 2, 3 and 5

      We illustrate the complexity of Ω m ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq78_HTML.gif, the algebraic polynomial linking the input and output as we move from u = z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq79_HTML.gif to v = m z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq80_HTML.gif, by first considering m = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq81_HTML.gif and m = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq82_HTML.gif.

      Example 6 ( λ ( 2 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq83_HTML.gif and λ ( 3 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq84_HTML.gif)

      As described in [1],
      λ ( 2 z ) 1 / 2 = 2 3 / 4 ϑ 4 4 ( z , e π ) ϑ 1 4 ( z , e π ) ϑ 3 4 ( z , e π ) + ϑ 1 4 ( z , e π ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ37_HTML.gif
      (37)
      and so letting τ ( z ) = ϑ 1 2 ( z , e π ) / ϑ 3 2 ( z , e π ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq85_HTML.gif, we obtain
      τ ( z ) = 2 λ ( z ) 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ38_HTML.gif
      (38)
      λ ( 2 z ) = 2 3 / 2 ( λ 2 ( z ) τ 2 ( z ) 1 + τ 2 ( z ) ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ39_HTML.gif
      (39)
      Iteration yields Ω 2 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq86_HTML.gif. From this one may recursively compute λ ( z / 2 n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq87_HTML.gif from λ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq88_HTML.gif and watch the tower of radicals grow. Correspondingly,
      λ ( 3 z ) = 2 ( λ ( 2 z ) λ ( z ) τ ( 2 z ) τ ( z ) ) 2 ( 1 + τ ( 2 z ) τ ( z ) ) 2 λ ( z ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ40_HTML.gif
      (40)
      where τ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq89_HTML.gif, τ ( 2 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq90_HTML.gif, λ ( 2 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq83_HTML.gif are given by (38) and (39). After simplification, we obtain
      λ ( 3 z ) = λ ( z ) ( λ 4 ( z ) 6 λ 2 ( z ) + 6 2 λ ( z ) 3 3 λ 4 ( z ) 6 2 λ 3 ( z ) + 6 λ 2 ( z ) 1 ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ41_HTML.gif
      (41)

      In addition, since we have an algebraic relation α = 1 λ / 2 1 1 / λ / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq91_HTML.gif, this allows us computationally to prove the evaluation of ϕ 2 ( x / 2 , y / 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq92_HTML.gif and ϕ 2 ( x / 3 , y / 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq93_HTML.gif once ϕ 2 ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq5_HTML.gif is determined as it is for the cases of Theorem 1.

      We may perform the same work inductively for d = 2 n + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq94_HTML.gif using ( n + 1 ) z ± n z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq95_HTML.gif in the known addition formulas for ϑ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq96_HTML.gif [1] to obtain Ω 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq97_HTML.gif and so on. The inductive step is
      λ ( ( 2 n + 1 ) z ) = 2 λ ( z ) ( λ ( ( n + 1 ) z ) λ ( n z ) τ ( ( n + 1 ) z ) τ ( n z ) 1 + τ ( ( n + 1 ) z ) τ ( n z ) ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ42_HTML.gif
      (42)

      Example 7 (Empirical computation of Ω d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq98_HTML.gif)

      Given z and u = λ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq99_HTML.gif, v = λ ( d z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq100_HTML.gif, we compute u j v k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq101_HTML.gif for 0 j J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq102_HTML.gif, 0 k K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq103_HTML.gif up to degree J, K and look for a relation to precision D. This is a potential candidate for Ω d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq98_HTML.gif. If Ω d ( λ ( w ) , λ ( d w ) ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq104_HTML.gif at a precision significantly greater than D and for various choices of w, we can reliably determine Ω d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq98_HTML.gif in this way [1].

      Example 8

      We conclude this subsection by proving the following empirical evaluation:
      ϕ 2 ( 1 / 5 , 2 / 5 ) = ? 1 4 π log 5 1 / 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ43_HTML.gif
      (43)
      Let w = π 10 ( 1 + 2 i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq105_HTML.gif so that
      ϕ 2 ( 2 / 5 , 1 / 5 ) = log | α ( w ) | 4 π while  ϕ 2 ( 2 / 5 , 1 / 5 ) = log | α ( 2 w ) | 4 π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Eque_HTML.gif
      since ϕ 2 ( 2 / 5 , 4 / 5 ) = ϕ 2 ( 2 / 5 , 1 / 5 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq106_HTML.gif. We may - with help from a computer algebra system - solve for the stronger requirement that α ( 2 w ) = α ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq107_HTML.gif using Example 6. We obtain
      α ( 2 w ) = 2 i 1 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equf_HTML.gif

      so that | α ( 2 w ) | = 5 1 / 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq108_HTML.gif and | α ( w ) | = 5 1 / 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq109_HTML.gif, as required. To convert this into a proof, we can make an a priori estimate of the degree and length of α ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq110_HTML.gif - using (39), (41) and (42) while λ ( 5 w ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq111_HTML.gif - and then perform a high precision computation to show no other algebraic number could approximate the answer well enough. The underlying result we appeal to [[6], Exercise 8, p.356] is given next in Theorem 9.

      In this particular case, we use P ( α ) = α 4 + 2 α 2 + 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq112_HTML.gif with = 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq113_HTML.gif, d = 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq114_HTML.gif and need to confirm that | P ( α ) | < 5 D / 4 L 3 8 D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq115_HTML.gif. A very generous estimate of L < 10 12 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq116_HTML.gif and D < 10 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq117_HTML.gif shows it is enough to check | P ( α ( w ) ) | 10 765 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq118_HTML.gif. This is very easy to confirm. Relaxing to L < 10 100 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq119_HTML.gif, D < 10 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq120_HTML.gif requires verifying | P ( α ( w ) ) | 10 7 , 584 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq121_HTML.gif. This takes only a little longer.

      Recall that the length of a polynomial is the sum of the absolute value of the coefficients.

      Theorem 9 (Determining a zero)

      Suppose P is an integral polynomial of degree D and length L. Suppose that α is algebraic of degree d and length . Then either P ( α ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq122_HTML.gif or
      | P ( α ) | max { 1 , | α | } D L d 1 D . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equg_HTML.gif

      Example 10 ( λ ( 5 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq123_HTML.gif)

      Making explicit the recipe for λ ( 5 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq123_HTML.gif, we eventually arrive at
      λ ( 5 z ) = λ ( z ) ( λ 4 ( z ) 4 2 λ 3 ( z ) + 14 λ 2 ( z ) 10 2 λ ( z ) + 5 ) 2 ( 5 λ 4 ( z ) 10 2 λ 3 ( z ) + 14 λ 2 ( z ) 4 2 λ ( z ) + 1 ) 2 × ( λ 8 ( z ) + 4 2 λ 7 ( z ) 32 λ 6 ( z ) + 36 2 λ 5 ( z ) 34 λ 4 ( z ) + 4 2 λ 3 ( z ) + 8 λ 2 ( z ) 4 2 λ ( z ) + 1 ) 2 ( λ 8 ( z ) 4 2 λ 7 ( z ) + 8 λ 6 ( z ) + 4 2 λ 5 ( z ) 34 λ 4 ( z ) + 36 2 λ 3 ( z ) 32 λ 2 ( z ) + 4 2 λ ( z ) + 1 ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ44_HTML.gif
      (44)

      This shows how generous our estimates were in Example 8.

      In similar fashion we can now computationally confirm all of the exact evaluations in Conjecture 2 and the like. For example, we know that | α ( π ( 1 / 3 + i / 3 ) ) | = | 1 / α ( π / 2 ( 2 / 3 + 2 i / 3 ) ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq124_HTML.gif. This solves to produce α ( π / 2 ( 1 / 3 + i / 3 ) ) 2 = 1 + 2 / 3 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq125_HTML.gif and establishes (16) of Theorem 1. Now Example 6 can be used to produce α ( π / 2 ( 1 / 6 + i / 6 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq126_HTML.gif is as given in Conjecture 2, and so on.

      We now turn to our main focus.

      3.5 ‘Compressed’ potentials

      Yet another solution to the Poisson equation with crystal charge source, for d > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq1_HTML.gif, is
      ϕ 2 ( x , y , d ) : = 1 π 2 m , n O 2 cos ( π m x ) cos ( π n d y ) m 2 + d n 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ45_HTML.gif
      (45)

      This is the potential inside a crystal compressed by 1 / d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq127_HTML.gif on the y-axis, in the sense that the Delord-cube now becomes the cuboid { x , y } [ 1 / 2 , 1 / 2 ] × [ 1 / ( 2 d ) , 1 / ( 2 d ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq128_HTML.gif. Indeed, ϕ 2 ( x , y , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq129_HTML.gif vanishes on the faces of this 2-cuboid (rectangle) for d > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq1_HTML.gif and integer. This is technically only a compression when d > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq130_HTML.gif.

      Along the same lines as the analysis of (13), we can posit a fast series
      ϕ 2 ( x , y , d ) = 1 π d R O + sinh ( π R d ( 1 / 2 | x | ) ) cos ( π R d y ) R cosh ( π R d / 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ46_HTML.gif
      (46)

      valid on the Delord cuboid, i.e., x [ 1 / 2 , 1 / 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq131_HTML.gif, y [ 1 / ( 2 d ) , 1 / ( 2 d ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq132_HTML.gif. Moreover, the log-accumulation technique of [[5], Appendix] applied as with Theorem 4 yields the following.

      Theorem 11 (ϑ-representation for compressed potential)

      We have
      ϕ 2 ( x , y , d ) = 1 2 d π log | ϑ 2 ( z , q ) ϑ 4 ( z , q ) ϑ 1 ( z , q ) ϑ 3 ( z , q ) | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ47_HTML.gif
      (47)

      where q : = exp ( π d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq133_HTML.gif and z : = 1 2 π d ( y + i x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq134_HTML.gif.

      This theorem is consistent with d = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq135_HTML.gif in the sense ϕ 2 ( x , y , 1 ) = ϕ 2 ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq136_HTML.gif. In addition, for integers a , b 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq137_HTML.gif, we have
      a 2 ϕ 2 ( a x , a y , b 2 a 2 ) : = 1 π 2 m a O , n b O cos ( π m x ) cos ( π n y ) m 2 + n 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ48_HTML.gif
      (48)
      We note that there are various trivial evaluations such as ϕ 2 ( x , 1 / 4 , 4 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq138_HTML.gif for all x. Moreover, for each positive rational d, there is an analogue of Theorem 5 in which 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq139_HTML.gif is replaced by the d th singular value k d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq140_HTML.gif [2, 6]. Thence, k 2 = 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq141_HTML.gif, k 3 = ( 3 1 ) / 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq142_HTML.gif, and k 4 = ( 2 1 ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq143_HTML.gif. Precisely, we set
      λ d ( z ) : = ϑ 4 2 ( z , exp ( π d ) ) ϑ 3 2 ( z , exp ( π d ) ) , κ d ( z ) : = ϑ 2 2 ( z , exp ( π d ) ) ϑ 3 2 ( z , exp ( π d ) ) , τ d ( z ) : = ϑ 1 2 ( z , exp ( π d ) ) ϑ 3 2 ( z , exp ( π d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ49_HTML.gif
      (49)
      and have (where k d = 1 k d 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq144_HTML.gif)
      k d λ d ( z ) + k d κ d ( z ) = 1 , k d λ d ( z ) k d κ d ( z ) = τ d ( z ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ50_HTML.gif
      (50)
      (see [[6], Prop. 2.1]). We observe that (47) can be written for all x, y as
      ϕ 2 ( x , y , d ) = 1 4 d π log | 1 k d λ d ( z ) 1 k d / λ d ( z ) | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ51_HTML.gif
      (51)

      on appealing to (49) and (50). Hence we may now study λ d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq145_HTML.gif exclusively.

      Fix an integer m > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq146_HTML.gif. The addition formulas for the ϑ’s as given in [[6], Section 2.6]
      ϑ 3 ( z + w , q ) ϑ 3 ( z w , q ) ϑ 3 2 ( 0 , q ) = ϑ 3 2 ( z , q ) ϑ 3 2 ( w , q ) + ϑ 1 2 ( z , q ) ϑ 1 2 ( w , q ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ52_HTML.gif
      (52)
      ϑ 4 ( z + w , q ) ϑ 4 ( z w , q ) ϑ 4 2 ( 0 , q ) = ϑ 4 2 ( z , q ) ϑ 4 2 ( w , q ) ϑ 1 2 ( z , q ) ϑ 1 2 ( w , q ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ53_HTML.gif
      (53)
      ϑ 1 ( z + w , q ) ϑ 1 ( z w , q ) ϑ 2 ( 0 , q ) ϑ 3 ( 0 , q ) = ϑ 1 ( z , q ) ϑ 4 ( z , q ) ϑ 2 ( w , q ) ϑ 3 ( w , q ) + ϑ 1 ( w , q ) ϑ 4 ( w , q ) ϑ 2 ( z , q ) ϑ 3 ( z , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ54_HTML.gif
      (54)

      when replacing z by ( m 1 ) z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq147_HTML.gif and w by z, and appealing to (50), allow one recursively to write λ d ( m z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq148_HTML.gif algebraically in terms of λ d ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq149_HTML.gif. We first give the equation for λ d ( 2 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq150_HTML.gif.

      Example 12 ( λ d ( 2 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq150_HTML.gif)

      As for λ ( 2 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq83_HTML.gif, which is the case d = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq41_HTML.gif, we have analytically
      τ d ( z ) = λ d ( z ) k d k d , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ55_HTML.gif
      (55)
      λ d ( 2 z ) = k d 3 ( λ d 2 ( z ) τ d 2 ( z ) 1 + τ d 2 ( z ) ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ56_HTML.gif
      (56)
      This becomes
      λ d ( 2 z ) = k d ( 1 + λ d 2 ( z ) 2 λ d ( z ) / k d 1 + λ d 2 ( z ) 2 λ d ( z ) k d ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ57_HTML.gif
      (57)

      and λ d ( 0 ) = k d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq151_HTML.gif.

      We turn to λ d ( 3 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq152_HTML.gif.

      Example 13 ( λ d ( 3 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq152_HTML.gif)

      As for λ ( 3 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq84_HTML.gif, which is the case d = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq41_HTML.gif, we have
      τ d ( z ) = λ d ( z ) k d k d , τ d ( 2 z ) = λ d ( 2 z ) k d k d , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ58_HTML.gif
      (58)
      λ d ( 3 z ) = 1 k d 2 λ d ( z ) ( λ d ( z ) λ d ( 2 z ) τ d ( z ) τ d ( 2 z ) 1 + τ d ( z ) τ d ( 2 z ) ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ59_HTML.gif
      (59)
      In tandem with (57) this becomes
      λ d ( 3 z ) = λ d ( 4 λ d 3 k d 6 λ d 3 k d + λ d 4 k d + 4 λ d k d 2 ) 2 ( 4 λ d 3 + k d 6 λ d 2 k d 3 λ d 4 k d + 4 λ d 3 k d 2 ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ60_HTML.gif
      (60)

      since λ d ( 0 ) = k d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq151_HTML.gif.

      This can also be determined by using the methods of Example 7 to obtain the algebraic equation linking u = λ d ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq153_HTML.gif and v = λ d ( 3 z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq154_HTML.gif and then carefully factoring the result - guided by results such as Example 10. It could also have been rigorously proven by the method used for corresponding formula with d = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq41_HTML.gif in [1].

      We next exhibit several sample compressed-sum evaluations, of which (62) is the most striking.

      Theorem 14 (Some explicit compressed sums)

      ϕ 2 ( 1 3 , 2 3 , 2 ) = 1 8 2 π log ( 9 6 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ61_HTML.gif
      (61)
      ϕ 2 ( 1 3 , 3 3 , 3 ) = 1 8 3 π log ( 1 3 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ62_HTML.gif
      (62)
      ϕ 2 ( 1 3 , 1 3 , 4 ) = 1 16 π log ( 3 ( 2 + 3 ) ( 2 3 ) 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ63_HTML.gif
      (63)
      ϕ 2 ( 1 4 , 1 4 , 9 ) = 1 48 π log ( 1 4 ( 3 + 5 3 4 2 ) ( 3 1 ) 3 ( 1 + 2 ) 2 ( 3 2 ) 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ64_HTML.gif
      (64)
      Proof Let us first sketch a proof of (62). This requires showing α = exp ( 8 π 3 ψ 2 ( 1 3 , 1 / 3 , d ) ) = 1 / 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq155_HTML.gif. Direct computer algebra using (51) shows that it suffices to prove that λ 3 ( z ) = 1 / 2 + 1 / 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq156_HTML.gif when z 3 : = π ( 1 / 2 + i 3 / 6 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq157_HTML.gif. Now (60) becomes
      λ 3 ( 3 z ) = λ 3 ( 4 λ 3 3 k 3 6 λ 3 2 k 3 + λ 3 4 k 3 + 4 λ 3 k 3 2 ) 2 ( 4 λ 3 3 + k 3 6 λ 3 2 k 3 3 λ 3 4 k 3 + 4 λ 3 3 k 3 2 ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ65_HTML.gif
      (65)
      However, 3z is in the compressed lattice generated by 1, i / 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq158_HTML.gif and so
      λ 3 ( 3 z ) { 0 , , k 3 = 3 + 1 2 2 , λ 3 ( i π 3 2 ) = 3 2 3 1 2 2 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equh_HTML.gif

      From this, we may deduce that λ 3 ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq159_HTML.gif is a zero of the denominator of (65), whose only real root is 1 / 2 + 1 / 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq160_HTML.gif.

      To prove (61), we similarly consider (51) and (60) with d = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq161_HTML.gif so that k 2 = 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq141_HTML.gif and z 2 : = π ( 1 + i / 2 ) / 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq162_HTML.gif is in the compressed lattice. Indeed, λ 2 ( 3 z ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq163_HTML.gif, and we now deduce that
      λ 2 ( z ) = 2 + 2 2 + i 10 2 14 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equi_HTML.gif

      and the result again follows.

      To prove (63) requires an application of this method with d = 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq114_HTML.gif and x = 1 / 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq164_HTML.gif and y = 1 / 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq165_HTML.gif. We likewise may prove (64) (with d = 9 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq166_HTML.gif). In this final case, x = 1 / 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq167_HTML.gif and y = 1 / 12 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq168_HTML.gif. In these cases, we need to appeal to both (59) and (60) at least once. □

      In general, the polynomial for α = exp ( 8 π d ψ 2 ( x , y d , d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq169_HTML.gif with x, y rational can be calculated as we describe in the next section.

      3.6 Further results for ϕ 2 ( a / c , b / c d , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq170_HTML.gif

      As noted, we anticipated similarly algebraic evaluations for ϕ 2 ( a / c , b / c d , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq171_HTML.gif for nonnegative integer values of a, b, c, d. And, as we will show in more examples in the next section, this is indeed true. All such results can, in principle, be rigorously established by the techniques of Example 7 and Example 8 or indeed Theorem 14, but we will not do this. We do however give the first few polynomials, P d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq172_HTML.gif, for exp ( 8 π ϕ 2 ( 1 / d , 1 / d , d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq173_HTML.gif, that is, with a = b = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq174_HTML.gif, c = d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq175_HTML.gif.

      Example 15 (Some explicit compressed polynomials for exp ( 8 π d ϕ 2 ( 1 / d , 1 / d , d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq176_HTML.gif)

      We present here the polynomials P d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq172_HTML.gif when 3 d 9 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq177_HTML.gif.

      d

      minimal polynomial

      3

      1 − 3α

      4

      1 − 644α + 3 , 334 α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq178_HTML.gif 644 α 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq179_HTML.gif + α 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq180_HTML.gif

      5

      1 + 20α 210 α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq181_HTML.gif + 100 α 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq182_HTML.gif + 25 α 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq183_HTML.gif

      6

      1 − 39 , 368 α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq184_HTML.gif + 309 , 916 α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq185_HTML.gif + 589 , 192 α 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq186_HTML.gif + 639 , 814 α 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq187_HTML.gif + 589 , 192 α 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq188_HTML.gif + 309 , 916 α 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq189_HTML.gif 39 , 368 α 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq190_HTML.gif + α 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq191_HTML.gif

      7

      1 − 14α + 21 α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq192_HTML.gif 7 α 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq193_HTML.gif

      8

      1 − 972 , 816 α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq194_HTML.gif + 108 , 957 , 816 α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq195_HTML.gif 3 , 146 , 762 , 800 α 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq196_HTML.gif + 87 , 457 , 048 , 348 α 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq197_HTML.gif 579 , 962 , 120 , 464 α 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq198_HTML.gif 2 , 624 , 405 , 729 , 464 α 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq199_HTML.gif 52 , 677 , 445 , 809 , 328 α 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq200_HTML.gif + 115 , 992 , 837 , 288 , 518 α 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq201_HTML.gif 52 , 677 , 445 , 809 , 328 α 9 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq202_HTML.gif 2 , 624 , 405 , 729 , 464 α 10 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq203_HTML.gif 579 , 962 , 120 , 464 α 11 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq204_HTML.gif + 87 , 457 , 048 , 348 α 12 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq205_HTML.gif 3 , 146 , 762 , 800 α 13 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq206_HTML.gif + 108 , 957 , 816 α 14 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq207_HTML.gif 972 , 816 α 15 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq208_HTML.gif + α 16 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq209_HTML.gif

      9

      1 + 438α 8 , 997 α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq210_HTML.gif + 8 , 292 α 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq211_HTML.gif + 711 α 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq212_HTML.gif 954 α 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq213_HTML.gif 3 α 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq214_HTML.gif

      Note that reversing the order of the variables leads to a more intractable computation since ϕ 2 ( 1 / d , 1 / d , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq215_HTML.gif is not of the form covered by Theorem 11.

      3.7 Computational results

      Results that we have obtained for the special set of cases exp ( 8 π d ϕ 2 ( 1 / d , 1 / d , d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq216_HTML.gif, for integers d up to 40, are shown in Table 1. Our computations required up to 25,000-digit precision, and, for large degrees and correspondingly high precision levels, were rather expensive (over 20 processor-hours in some cases). We employed the ARPREC arbitrary precision software [11].
      Table 1

      PSLQ runs to recover minimal polynomials satisfied by exp ( 8 π d ϕ 2 ( 1 / d , 1 / d , d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq217_HTML.gif

      d

      m(d)

      P

      T

      log 10 M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq218_HTML.gif

      log 10 M m ( d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq219_HTML.gif

      3

      1

      200

      0.00

        

      4

      4

      200

      0.01

      3.5230

      0.8807

      5

      4

      200

      0.01

      2.3222

      0.5806

      6

      8

      200

      0.02

      5.8061

      0.7258

      7

      3

      200

      0.01

      1.1461

      0.3820

      8

      16

      1,000

      0.85

      14.0644

      0.8790

      9

      6

      200

      0.01

      3.9187

      0.6531

      10

      16

      1,000

      0.76

      12.1262

      0.7579

      11

      15

      1,000

      0.55

      7.5230

      0.5015

      12

      16

      1,000

      0.85

      15.2686

      0.9543

      13

      12

      1,000

      0.45

      9.5690

      0.7974

      14

      48

      4,000

      100.31

      38.4539

      0.8011

      15

      8

      200

      0.03

      2.1139

      0.2642

      16

      32

      2,000

      12.51

      33.7985

      1.0562

      17

      32

      2,000

      8.93

      21.9952

      0.6874

      18

      24

      2,000

      4.32

      21.1366

      0.8807

      19

      27

      2,000

      5.27

      16.8591

      0.6244

      20

      64

      5,000

      415.78

      58.8250

      0.9191

      21

      24

      2,000

      4.52

      15.6374

      0.6516

      22

      40

      3,000

      64.33

      41.4566

      1.0364

      23

      33

      3,000

      10.92

      14.4705

      0.4385

      24

      64

      5,000

      412.59

      60.3300

      0.9427

      25

      20

      2,000

      3.77

      20.0766

      1.0038

      26

      144

      25,000

      71,680.30

      121.91

      0.8466

      27

      27

      2,000

      5.70

      18.9234

      0.7009

      28

      48

      4,000

      131.29

      58.4901

      1.2185

      29

      84

      6,000

      1,375.38

      60.0921

      0.7154

      30

      64

      5,000

      557.29

      56.9952

      0.8906

      31

      45

      4,000

      38.89

      19.8425

      0.4409

      33

      40

      4,000

      81.42

      32.1363

      0.8034

      34

      128

      21,300

      45,993.71

      123.9012

      0.9680

      35

      72

      12,000

      1,179.43

      41.3569

      0.5744

      36

      96

      12,000

      95.3311

      95.3311

      0.9930

      37

      36

      3,000

      29.11

      43.5933

      1.2110

      39

      48

      4,000

      56.33

      20.7849

      0.4330

      40

      128

      21,300

       

      127.3572

      0.9950

      Here m(d) is the degree, P is the precision level in digits, T is the run time in seconds, and log10M is the size in digits of the central coefficient. Degrees in bold were obtained by Andrew Mattingly.

      The degree m ( d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq220_HTML.gif of the minimal polynomial for exp ( 8 π d ϕ 2 ( 1 / d , 1 / d , d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq221_HTML.gif, the number of zeroes z ( d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq222_HTML.gif among the minimal polynomial coefficients, the numeric precision level P, the run time in seconds T, and the approximate base-10 logarithm M of the absolute value of the central coefficient are also shown in the table, together with the ratio ( log 10 M ) / m ( d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq223_HTML.gif.

      We were not able to obtain relations for all cases d 40 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq224_HTML.gif. Evidently, the precision levels we employed in these cases ( d = 32 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq225_HTML.gif and d = 38 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq226_HTML.gif) were still insufficient to recover the underlying relations; a method to obtain the result for d = 32 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq227_HTML.gif is described in Remark 16 - which works well because 32 = 4 × 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq228_HTML.gif. It is interesting that in this work, as opposed to the ϕ ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq229_HTML.gif constants we studied in [1], odd values and prime values of d generally yielded simpler relations than the even instances.

      In the earlier study [1], we included Jason Kimberley’s observation that the degree m ( d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq220_HTML.gif of the minimal polynomial for exp ( 8 π ϕ 2 ( 1 / d , 1 / d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq230_HTML.gif appeared to be given for odd primes by m ( 4 k + 1 ) = ( 2 k ) ( 2 k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq231_HTML.gif and m ( 4 k + 3 ) = ( 2 k + 2 ) ( 2 k + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq232_HTML.gif. If one sets m ( 2 ) = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq233_HTML.gif, for notational convenience, then it appears that for any prime factorization of an integer greater than 2,
      m ( i = 1 k p i e i ) = ? 4 k 1 i = 1 k p i 2 ( e i 1 ) m ( p i ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ66_HTML.gif
      (66)

      Unfortunately, we have not yet been able to find a similar formula for the minimal polynomial degree in the compressed cases under study in Table 1. We are actively investigating this at the present time.

      With regards to the run times listed in Table 1 (given to 0.01 second accuracy), it should be recognized that like all computer run times, particularly in a multicore or multiprocessor environment, they are only repeatable to two or three significant digits. They are listed here only to emphasize the extremely rapid increase in computational cost as the degree m and corresponding precision level P increase.

      Remark 16 (A refined approach)

      A referee has suggested making more more explicit the relations with elliptic functions. We had opted to limit the theory needed so as to better understand what experimental computation could produce for a wider audience.
      1. (a)
        The basic quantity λ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq71_HTML.gif, which appears in Theorem 4, can be written in terms of the Jacobian elliptic function d n ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq234_HTML.gif [10]. If z = π ( y + i x ) / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq235_HTML.gif, and k = k 1 = 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq236_HTML.gif as above, then
        λ ( z ) = k d n 2 ( y K + i x K ; k 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equj_HTML.gif

        Notice that K = K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq237_HTML.gif in this case. All multiplication formulas for λ are then essentially special cases of addition formulas for Jacobian elliptic functions. At least in outline this makes it clear that λ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq71_HTML.gif is algebraic, since elliptic functions evaluated at rational multiples of the periods are always algebraic with respect to k. In [12] some closely related lattice sums (essentially linear combinations of some of the lattice sums being considered herein) are evaluated in part by such reasoning.

         
      2. (b)
        Similarly, for fixed 0 < k < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq238_HTML.gif, set
        f ( x ) : = 1 c n ( x ; k 2 ) 1 + c n ( x ; k 2 ) , g ( u ) : = 4 u ( 1 + u ) 2 4 k 2 u ( 1 u 2 ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equk_HTML.gif
         
      Then, in terms of the elliptic function cn [6, 10], we can write
      ϕ 2 ( x , y , d ) = 1 4 π d log | f ( 2 i K d / d , k d ) | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ67_HTML.gif
      (67)
      and the corresponding duplication formula for f is f ( 2 x ) = g ( f ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq239_HTML.gif. Now the value of k 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq240_HTML.gif and the Landen transform [6] yield
      k 32 = 1 1 ( 5 + 4 2 2 14 + 10 2 ) 2 1 + 1 ( 5 + 4 2 2 14 + 10 2 ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equl_HTML.gif
      [[2], Appendix] which makes the minimal polynomial for k 32 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq241_HTML.gif easily expressible
      x 8 1 , 800 x 7 12 , 772 x 6 63 , 800 x 5 105 , 402 x 4 63 , 800 x 3 12 , 772 x 2 1 , 800 x + 1 = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_Equ68_HTML.gif
      (68)

      Let θ 32 : = f ( 2 i K 32 / 32 , k 32 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq242_HTML.gif. Observe, much as in earlier examples, that g ( g ( g ( g ( θ 32 ) ) ) ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq243_HTML.gif. We may now obtain an algebraic equation relating k 32 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq241_HTML.gif, θ 32 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq244_HTML.gif by computing g ( g ( g ( g ( u ) ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq245_HTML.gif while reducing repeatedly using (68); finally, resolvant computations produce the equation for d = 32 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq227_HTML.gif that we failed to find in Table 1.

      4 Conclusions

      In this study, we have obtained some explicit closed-form evaluations of certain summations that arise as solutions of the Poisson equation, which in turn arises in numerous arenas ranging from engineering to studies of crystal structures. Many of these evaluations were first found experimentally by means of a process we have employed in other settings: computing the relevant expressions to high precision, then applying the PSLQ algorithm to recognize the resulting numerical values in closed form. In this case, we computed the constants exp ( 8 π d ϕ 2 ( 1 / d , 1 / d , d ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-75/MediaObjects/13661_2012_Article_333_IEq246_HTML.gif, for various integers d, to a numeric precision of up to 25,000 digits, then found the integer coefficients of a polynomial that is satisfied by the constant up to a tolerance corresponding to the precision level. We then demonstrated a theoretical framework whereby such experimentally discovered relations can, in many cases, be formally proven.

      For further details of the computation of lattice sums and related theta functions the reader is referred to [1322].

      Declarations

      Acknowledgements

      Dedicated to Professor Hari M Srivastava.

      The authors thank many colleagues and the referees for fruitful discussions about lattice sums and theta functions. The first author supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. The second author is supported in part by the Australian Research Council.

      Authors’ Affiliations

      (1)
      Lawrence Berkeley National Laboratory
      (2)
      Centre for Computer Assisted Research Mathematics and Its Applications (CARMA), University of Newcastle

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      Copyright

      © Bailey and Borwein; licensee Springer. 2013

      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.