Lattice Sums Arising from the Poisson Equation
David H Bailey, Lawrence Berkeley National Lab, USA
Collaborators: Jonathan M. Borwein (Univ. of Newcastle, Australia),
Richard Crandall (Reed College) and I. J. Zucker (King’s College, London)
This talk is available at:
http://www.davidhbailey.com/dhbtalks/dhb-lattice.pdf

1Lattice sum solutions to the Poisson
equation
The Poisson equation for an n-dimensional point-charge source gives rise toan electric static potential of the form

2nn (r) = 2⇡X

m1 ,...,mn > 0, oddcos(⇡m1 r1 ) · · · cos(⇡mn rn )m21 + · · · m2n

Sums of this type have been studied widely in mathematical physics andalso have applications in image processing (to sharpen images).
In two recent papers (see below), we found closed forms for these sums inmany cases, by taking advantage of connections with Jacobi theta
functions.

1. DHB, J. M. Borwein, R. E. Crandall and J. Zucker, “Lattice sums arising from the Poisson equation,”
manuscript, http://www.davidhbailey.com/dhbpapers/PoissonLattice.pdf
2. DHB and J. M. Borwein, “Compressed lattice sums arising from the Poisson equation: Dedicated toProfessor Hari Sirvastava,” manuscript, http://www.davidhbailey.com/dhbpapers/Poissond.pdf.
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Algebraic numbers in Poisson potentialfunctions associated with lattice sums
In particular, we discovered numerically, and then proved, that for rational(x, y), the two-dimensional Poisson potential function satisfies

12 (x, y) = 2⇡

Xm,n odd

1cos(m⇡x) cos(n⇡y)= log ↵22m +n⇡

where α is an algebraic number, i.e., the root of an integer polynomial:

0 = a0 + a1 ↵ + a2 ↵ 2 + · · · + an ↵ nNumerous other results, both particular and general, are presented in thetwo papers.

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Fast series for computing φn(x,y)
Our approach was to compute φn(x,y) to very high precision (hundreds or
thousands of digits), then apply the PSLQ algorithm to identify the computedvalues. We discovered that these are always of the form 1/π log (α).
These computations were enabled by finding the following very rapidly
convergent series for φn(x,y):1cosh(⇡x) + cos(⇡y)
log
2 (x, y) =4⇡
cosh(⇡x) cos(⇡y)

2⇡Xm>0 oddcosh(⇡mx) cos(⇡my)
m(1 + e⇡m )The minimal polynomials for α = exp (π φ2(x,y)) were found by PSLQ
calculations, with the (n+1)-long vector (1, α, α2, …, αn) as input.
PSLQ returns the vector of integer coefficients (a0, a1, a2, …, an) as output.

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The PSLQ integer relation algorithm
Let (xn) be a given vector of real numbers. An integer relation algorithm
finds integers (an) such thata1 x1 + a2 x2 + · · · + an xn= 0(or within “epsilon” of zero, where epsilon = 10-p and p is the precision).
At the present time the “PSLQ” algorithm of mathematician-sculptor
Helaman Ferguson is the most widely used integer relation algorithm. It
was named one of ten “algorithms of the century” by Computing in Scienceand Engineering.
Integer relation detection requires very high precision (at least n*d digits,
where d is the size in digits of the largest ak), both in the input data and in
the operation of the algorithm.
1. H. R. P. Ferguson, DHB and S. Arno, “Analysis of PSLQ, An Integer Relation Finding Algorithm,”
Mathematics of Computation, vol. 68, no. 225 (Jan 1999), pg. 351-369.
2. DHB and D. J. Broadhurst, “Parallel Integer Relation Detection: Techniques and Applications,”
Mathematics of Computation, vol. 70, no. 236 (Oct 2000), pg. 1719-1736.
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PSLQ, continued· PSLQ constructs a sequence of integer-valued matrices Bn that reduces
the vector y = x * Bn, until either the relation is found (as one of thecolumns of Bn), or else precision is exhausted.
· At the same time, PSLQ generates a steadily growing bound on the sizeof any possible relation.
· When a relation is found, the size of smallest entry of the y vector
suddenly drops to roughly “epsilon” (i.e. 10-p, where p is the number ofdigits of precision).
· The size of this drop can be viewed as a “confidence level” that therelation is real and not merely a numerical artifact -- a drop of 20+ ordersof magnitude almost always indicates a real relation.
Several efficient variants of PSLQ are available:· 2-level and 3-level PSLQ: performs almost all PSLQ iterations with only
double precision, updating full-precision arrays as needed. Hundreds oftimes faster than the original full-precision PSLQ algorithm.
· Multi-pair PSLQ: dramatically reduces the number of iterations required.
Designed for parallel system, but runs faster even on 1 CPU.
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Decrease of log10(min |yi|) in a multipair
PSLQ run7

Samples of minimal polynomials
discovered by PSLQ computations
k5
67
8910

Minimal polynomial for exp (8 π φ2(1/k,1/k))
1 + 52↵ 26↵2 12↵3 + ↵4
1 28↵ + 6↵2 28↵3 + ↵4
1 196↵ + 1302↵2 14756↵3 + 15673↵4 + 42168↵5 111916↵6 + 82264↵735231↵8 + 19852↵9 2954↵10 308↵11 + 7↵12
1 88↵ + 92↵2 872↵3 + 1990↵4 872↵5 + 92↵6 88↵7 + ↵8
1 534↵ + 10923↵2 342864↵3 + 2304684↵4 7820712↵5 + 13729068↵622321584↵7 + 39775986↵8 44431044↵9 + 19899882↵10 + 3546576↵118458020↵12 + 4009176↵13 273348↵14 + 121392↵1511385↵16 342↵17 + 3↵18
1 216↵ + 860↵2 744↵3 + 454↵4 744↵5 + 860↵6 216↵7 + ↵8The minimal polynomial for exp (8 π φ2(1/32,1/32)) has degree 128, withindividual coefficients ranging from 1 to over 1056. This PSLQ computation
required 10,000-digit precision. See next slide.
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Degree-128 minimal polynomial forexp (8 π φ2(1/32,1/32))

9Degrees of minimal polynomials foundfor exp(8 π φ2(1/d,1/d))
Legend:dArgumentm(d)Degree of minimal polyz(d)Zero coefficients
PPrecision in digits
TRun time in seconds
log10M Log10 of largest coeff.
Jason Kimberley noted thatm(d) satisfies:
mk
Y

i=1

pei i!

kY2(e 1)= 4k 1pi im(pi )
?

i=1

dm(d)

z(d)

P

T

log10 M

5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32

4
4
12
8
18
8
30
16
36
24
32
32
64
36
90
32
96
60
132
64
100
72
144
96
144
64
144
128

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
000
0

400
400
400
400
400
400
1000
400
1000
1000
1000
1000
4000
2000
6000
2000
6000
3000
12000
3000
8000
4000
12000
8000
12000
3000
12000
10000

0.40
0.39
0.71
0.43
1.81
0.54
28.50
1.22
44.04
12.37
34.58
29.62
3387.71274.6019559.37
222.8725210.51
1748.19212634.54
2224.4258723.90
4961.57
Failed46795.52
Failed2208.95
Failed163662.83

1.41500.7782
5.04893.29897.64772.6571
12.9873
6.788015.63859.724512.8370
13.8452
28.2396
13.8718
39.8456
13.970542.469625.800258.7280
28.162444.0690
30.961179.4540
42.5098
79.5125
27.2294
79.411956.8932

10000log10 Mm(d)0.35370.19450.42070.41240.42490.33210.43290.42430.43440.40520.40120.43270.44120.38530.44270.43660.44240.43000.44490.44000.44070.43000.55180.44280.55220.42550.55150.4445

For additional details
This talk is available at:
http://www.davidhbailey.com/dhbtalks/dhb-lattice.pdf
Full details are in given these two papers:
1. DHB, J. M. Borwein, R. E. Crandall and J. Zucker, “Lattice sums arisingfrom the Poisson equation,” manuscript, available at
http://www.davidhbailey.com/dhbpapers/PoissonLattice.pdf
2. DHB and J. M. Borwein, “Compressed lattice sums arising from thePoisson equation: Dedicated to Professor Hari Sirvastava,” manuscript,
http://www.davidhbailey.com/dhbpapers/Poissond.pdf

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