Computation and experimental evaluation ofMordell–Tornheim–Witten sum derivatives
David H. Bailey
http://www.davidhbailey.com
Lawrence Berkeley National Lab (retired)
University of California, Davis, Dept. of Computer Science
Collaborators:
Jonathan M. Borwein (deceased 2 Aug 2016), University of Newcastle, Australia
Jason Kimberley, University of Newcastle, Australia
Watson Ladd, University of California, Berkeley

November 30, 2016
1 / 21

The PSLQ integer relation algorithm
Let X = (xk ) be an (m + 1)-long real or complex vector. An integer relation algorithm
such as PSLQ finds a nontrivial integer vector A = (ak ) such thata0 x0 + a1 x1 + · · · + am xm = 0.
I The multipair PSLQ algorithm is a more efficient and parallelizable variant of

PSLQ, the most widely used integter relation algorithm (other researchers use avariant of the LLL algorithm).
I Integer relation detection (by any algorithm) requires very high precision: at least(m + 1) · maxk log10 |ak | digits, both in the input data and the algorithm.

I H. R. P. Ferguson, D. H. Bailey and S. Arno, “Analysis of PSLQ, an integer relation finding

algorithm,” Mathematics of Computation, vol. 68, no. 225 (Jan 1999), pg. 351–369.
I D. H. Bailey and D. J. Broadhurst, “Parallel integer relation detection: Techniques andapplications,” Mathematics of Computation, vol. 70, no. 236 (Oct 2000), 1719–1736.
2 / 21

Decrease of log 10(min |yi |) in multipair PSLQ run

3 / 21

Application of multipair PSLQ
One simple but important application of multipair PSLQ is to recognize a computed
numerical value as the root of an integer polynomial of degree m.
Example: The following constant is suspected to be an algebraic number:
α = 1.232688913061443445331472869611255647068988824547930576057634684778 . . .
What is its minimal polynomial?
Method: Compute the vector (1, α, α2 , · · · , αm ) for m = 30, then input this vector tomultipair PSLQ.
Answer (using 250-digit arithmetic):
0 = 697 − 1440α − 20520α2 − 98280α3 − 102060α4 − 1458α5 + 80α6 − 43920α7+ 538380α8 − 336420α9 + 1215α10 − 80α12 − 56160α13 − 135540α14 − 540α15+ 40α18 − 7380α19 + 135α20 − 10α24 − 18α25 + α30
4 / 21

The Poisson potential function
In 2012, Richard Crandall, while investigating techniques to sharpen images, notedthat each pixel was given by a form of the 2-D Poisson potential function:
1 X cos(mπx ) cos(nπy )φ2 (x , y ) = 2
π m,n odd
m2 + n2
In a 2013 study, we numerically discovered, and then proved the intriguing fact that for
rational (x , y ),1
φ2 (x , y ) = log απwhere α is algebraic, i.e., the root of a some integer polynomial of degree m.
By computing high-precision numerical values of φ2 (x , y ) for various specific rational x
and y , and applying a multipair PSLQ program, we were able to produce the explicit
minimal polynomials for α in numerous specific cases.
I D. H. Bailey, J. M. Borwein, R. E. Crandall and J. Zucker, “Lattice sums arising from the Poisson

equation,” Journal of Physics A: Mathematical and Theoretical, vol. 46 (2013), 115201.

5 / 21Samples of minimal polynomials found by multipair PSLQ
s
5
6
7
8
9

10

Minimal polynomial corresponding to x = y = 1/s:
1 + 52α − 26α2 − 12α3 + α4
1 − 28α + 6α2 − 28α3 + α4
−1 − 196α + 1302α2 − 14756α3 + 15673α4 + 42168α5 − 111916α6 + 82264α7−35231α8 + 19852α9 − 2954α10 − 308α11 + 7α12
1 − 88α + 92α2 − 872α3 + 1990α4 − 872α5 + 92α6 − 88α7 + α8
−1 − 534α + 10923α2 − 342864α3 + 2304684α4 − 7820712α5 + 13729068α6−22321584α7 + 39775986α8 − 44431044α9 + 19899882α10 + 3546576α11−8458020α12 + 4009176α13 − 273348α14 + 121392α15−11385α16 − 342α17 + 3α18
1 − 216α + 860α2 − 744α3 + 454α4 − 744α5 + 860α6 − 216α7 + α8These computations are very expensive. The case x = y = 1/32, for instance, required
10,000-digit arithmetic and ran for 45 hours. Other runs, using even higher precision,
ultimately failed, evidently due to subtle program bugs. Help!

6 / 21

Kimberley’s formula for the degree of the polynomial
Based on our preliminary results, Jason Kimberley of the University of Newcastle,
Australia observed that the degree m(s) of the minimal polynomial associated with thecase x = y = 1/s appears to be given by the following:
Set m(2) = 1/2. Otherwise for primes p congruent to 1 mod 4, set m(p) = int2 (p/2),
where int denotes greatest integer, and for primes p congruent to 3 mod 4, setm(p) = int (p/2)(int (p/2) + 1). Then for any other positive integer s whose prime
factorization is s = p1e1 p2e2 · · · prer ,?

r −1m(s) = 4rY
2(ei −1)pim(pi ).

i=1

Does Kimberley’s formula hold for larger s? Why?
What is the true mathematical connection between the pair of rationals (x , y ) and thealgebraic number α?
7 / 21

Three improvements to the Poisson polynomial computation program

1. MPFUN2015: A new thread-safe multiprecision package.
I

Speedup: 3X

2. A new 3-level multipair PSLQ program.
I

Speedup: 4.2X

3. Parallel implementation on a 16-core system.
I

Speedup: 12.2X

Overall speedup: 156X

8 / 21

−1−127936α1−119729568α2+86259615424α3−63414593420752α4−23761356230294720α5−21521006149475920864α6+611939502140287079872α7−368241444434585386649704α8+34534137997535622208800832α9+190487328667825240861480928α10+70731866698900460686569086656α11−30639321248717091789168538879600α12+1922209242084456061370070376570688α13−92225937246234731111918315818791008α14+5237085972458845348795423777776516544α15−235187168740813424983476512954481139732α16+7429223399159195923652315325576416851776α17−183888805409443551177519325744878574722592α18+3271185405201792963719366162166966763587520α19−30299858585212592953207274837044089721625936α20−359992131877582010695076712447360628122541504α21+21102025339314733846086441916632636325491195040α22−476414747523697562008419279360098519444810134848α23+7169682338305684610208768903330082567075749930088α24−77147038334570163492295981291240215533152479746240α25+576164594448741453064385715717563185909190291963744α26−2736845608824484767533372475869861068967680432337984α27+14259726273826198867906144815986191950918103600120720α28−418898418536517746458185180042147268305073686192184768α29+11576727630588055309375415098495616370526328495842993184α30−215864800528660839585002162871715414654518484882221347136α31+2990075428515164844805618172618994337159659201108555443326α32−32103405822691698883721869657536188329165809065032597536320α33+266563559238616545234045835785848715794979563412588549450912α34−1588318769762074127932309766905276050093368042694667151097024α35+4203983024865006644081516487096952711041894782888827508128720α36+45741795572596990194281792252600295928845217432176402329342656α37−888295322846459251495779433538199608179522746791003115766467360α38+9318868074863097497435367536227145321575550333899160360356887616α39−75360568030730663716613169159158032198635749984420852664531432312α40+511904470165868318936871843389229757000581029769096726546021748160α41−3023546278817700158319248501909046229060364677626955289951128897760α42+15762439699463291788277556458704672972787957041647009977980115608384α43−72736254087237112123433235647299134668422519990232924303557659578512α44+294655977143150502457165877012542972911987116693855121083332478262976α45−1023418666617067140855492177628022043641503497647927192890895515496096α46+2872077837131433597858520147550093715024860106120898508958083768560704α47−5244656757384067630568130031166724732476128944651080774800327815250116α48−4375176099824059357126327245955265067024929918868045662842672259365696α49+106846263928061290720995820179791755195031418846948198250527939644422688α50−723639781157123885224054475278025279616831219096501715148796353565656000α51+3859388734169523941501510320228124865152568559427283813598867448036672464α52−18252524309522593867333335673661626604394606878814374964432136569382704704α53+77801314121940709085266334072671847312436136451041737155601244216316832608α54−295924487894698517430363594146438746974018698578472622597771256623091267264α55+990883094386701201740101120896676716687418747972363663196338999871139087736α56−2890647897108739627601089028825831836951153432467718962671967596342224331584α57+7329280206578701233923536747523603867159738577214393279268225554697992024224α58−16529926761499923756105727653427646107756611125773189972311741253023019544512α59+36621401257350141587688985844214102107297318891842289523223570545646432107248α60−97806177157646206090474078202543089426709440278475346399840243186928760712768α61+342578593578669077931375617617876157438332517966938868387799492392630391951328α62−1318494126999138496197980784859753020673610723076044612133782854663456341738176α63+4770648039063802379141989053619421884321933507524681278243277292208295535890193α64−15472388710953109453763060955215809455238864792962040591046439233238251454944640α65+44993126133327412718394996675533962755303579158603905782013384675151509701621696α66−118993678023270686331414029749780909390537125214711043931901528442137124714048640α67+291344068959030996949609862561213601556661749401236700666696596438800158224020448α68−672232544304160020140857460151229781783982323755372371953704546852887742475693952α69+1483756546102154371991621621590988762258290163435879224071206411473630010421819712α70−3160698702072288781771809020419553277725945343183442108817290538424291765729105536α71+6501409535318122589047227869235783252522832570490016457500770280864513874652673648α72−12819483008841743167047686498254296089188806357669061135254105501043843161878338944α73+23940036962388573943673383046497610566473660443396196928270109685668366770350448320α74−41783013070841149166516454269070382975961508352312427230333483156459522180289158272α75+67368007706334464267463678880196649459723255934263643619439168617608299192825564576α76−99642265418236291841000783588241107943576928164973329187609122090423042369846900608α77+135684188062925983721829145899247186881926375911313411751554577363782314702542167104α78−175027359690635875118927274192015444148378662148930449526142186378361018115616958080α79+2306104703709760221799424330422404147474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192-degree minimal polynomial found by multipair PSLQ for x = y = 1/35

This polynomial has degree 192, with coefficients as large as 1085 . This computation
required 18,000-digit arithmetic and 34 CPU-hours.
The case (1/37, 1/37) required 51,000-digit arithmetic and 90 CPU-days (5.6 days on
a 16-core parallel system).
Kimberley’s formula was upheld for (1/s, 1/s), for all s up to 52 (except fors = 41, 43, 47, 49, 51, which were too expensive), and also for s = 60 and s = 64.
9 / 21

Palindromic polynomials
From our results, in the case (1/s, 1/s) where s is even, the resulting polynomial isalways palindromic (ak = am−k ). For instance, when s = 16,
p16 (α) = 1 − 1376α1 − 12560α2 − 3550496α3 + 81241720α4 − 169589984α5+ 1334964944α6 − 24307725984α7 + 238934926108α8 − 1043027124704α9+ 2328675366384α10 − 3219896325280α11 + 4238551472456α12− 10247414430048α13 + 28552105805904α14 − 55832851687968α15+ 70020268309062α16− 55832851687968α17 + 28552105805904α18 − 10247414430048α19+ 4238551472456α20 − 3219896325280α21 + 2328675366384α22− 1043027124704α23 + 238934926108α24 − 24307725984α25 + 1334964944α26− 169589984α27 + 81241720α28 − 3550496α29 − 12560α30 − 1376α31 + α32

Nitya Mani, an undergraduate student at Stanford University, observed that if α is aroot of a palindromic polynomial such as this, then α + 1/α is a root of a transformed
polynomial of half the degree. This fact can be used to significantly accelerate thecomputation of Poisson polynomials in the even case.
10 / 21

Proofs of Kimberley’s formula and the palindromic property
I On March 16, DHB presented our results at a seminar at the University ofCalifornia, Berkeley.
I Following the presentation, Watson Ladd, a graduate student in mathematics,

brought to our attention the fact that some of our conjectures should follow fromresults in the theory of elliptic curves, Gaussian integers and ideals.
I After some effort, Ladd produced proofs of Kimberley’s formula and thepalindromic property, which proofs were then incuded in our paper and returnedto the journal.
I The paper has now appeared:
David H. Bailey, Jonathan M. Borwein, Jason Kimberley and Watson Ladd,“Computer discovery and analysis of large Poisson polynomials,” Experimental
Mathematics, 27 Aug 2016.
I A preprint is available here:
http://www.davidhbailey.com/dhbpapers/poisson-res.pdf.

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Mordell–Tornheim–Witten sums
The simplest MTW sum is:
W (r , s, t) =

X
m,n≥1

1
mr ns (m + n)t.

Such sums arise in combinatorics, mathematical physics (e.g., Feynmann diagrams andstring theory), Lie algebras, number theory and numerous other fields. In special cases
these sums have simple evaluations. For example, when t = 0,
W (r , s, 0) =X 1 X 11=
= ζ(r )ζ(s).
mr ns
mr n≥1 ns
m≥1
m,n≥1
X

The n-dimensional MTW sum is defined for integer mi and positive real ri as
W (r1 , r2 , . . . , rn , t) =X1.
mr1 mr2 · · · mnrn (m1 + m2 + · · · + mn )t
m1 ,...,mn ≥1 1 2

Matsumoto proved that W can be continued meromorphically to the entire Cr +1 space.
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Special caseWe will focus on the special case r1 = r2 = · · · = rn = t = s for real s (analyticallycontinued as above), namelyX1
ωn+1 (s) =,(m1 m2 · · · mn (m1 + m2 + · · · + mn ))sm ,m ,...,m ≥1
1

2n

for n = 2, 3, · · · . These sums were studied by Tomkins, who conjectured that(−1)nωn+1 (0) =.
n+1This was proved by Romik for the case n = 2 and in the general case by Borwein andDilcher.
1. H. Tomkins, “An exploration of multiple zeta functions,” Honours Thesis, Dalhousie University,
21 Apr 2016.
2. J. M. Borwein and K. Dilcher, “Derivatives and fast evaluation of multiple Witten functions,”
manuscript, to appear in Ramanujan Journal, 2016.
3. D. Romik, “On the number of n-dimensional representations of SU(3), the Bernoulli numbers,
and the Witten zeta function,” 17 Apr 2015, https://arxiv.org/pdf/1503.03776v3.pdf.

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Key result from Tomkins’ thesis
Let r1 , r2 , . . . , rn , t be complex variables with ri ∈ N for 1 ≤ i ≤ n. Then for any realθ > 0,
Γ(t)W (r1 , r2 , . . . , rn , t) =Γ(t, (m1 + m2 + · · · + mn )θ)
r1 r2
m
m
· · · mnrn (m1 + m2 + · · · + mn )t
m ,m ,...m ≥1 1 2
X

1

2

n



+

X

n
θw Y

ua1 ,ua2 ,...,uak ≥0

w i=1

X

{a1 ,a2 ,...,ak }⊂{1,2,...,n}where w = t − (n − k) +Pk

j=1 (uaj − raj ) +Γ(1 − ri )kY(−1)uaj ζ(raj − uaj )j=1

uaj ! Γ(1 − raj )


,

Pn

i=1 ri .

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Example: Special case when n = 4ω4 (s) =+1  X Γ(s, (m + n + p)θ)Γ(s) m,n,p≥1 (mnp(m + n + p))s(−1)m+n+p ζ(s − m)ζ(s − n)ζ(s − p)θm+n+p+s
m!n!p!(m + n + p + s)
m,n,p≥0X+ 3Γ(1 − s)

X (−1)m+n ζ(s − m)ζ(s − n)θ m+n+2s−1m,n≥0+3(Γ(1 − s))2

m!n!(m + n + 2s − 1)

X (−1)p ζ(s − p)θ p+3s−2p≥0

p!(p + 3s − 2)

θ4s−3 + (Γ(1 − s))3,4s − 3where θ > 0 is an arbitrary real parameter, and s is real but not an integer.

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Computation of ωn derivatives
In initial computations of omega derivatives at zero, namely ωd0 (0) for d = 3, 4, . . .,Borwein and Dilcher found and then proved the intriguing experimental equivalenceω30 (0) = log(2π),
based purely on the numerical value of ω30 (0) as computed from a more complex
evaluation due to Romik, where the sum arises in counting representations of SU(3).
Tomkins then showed that
ω40 (0) = − log(2π) + ζ 0 (−2).
These results immediately raise the question of whether the higher-degree constants
ωd0 (0) have similarly elegant evaluations.

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Numerical issues
We computed these constants to 400-digit precision and then employed the multipair
PSLQ algorithm to attempt to obtain an analytic evaluation.
Straightforward evaluation is exceedingly expensive, since with each higher degree d,the summations involve one more level of loop nesting, and each higher level of loop
nesting typically increases the computational run time by a factor of 10 or more overthe previous level.
However, after carefully examining these formulas and the equivalent computer code,
we ultimately achieved a 30-million-fold speedup.

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PSLQ analysis
In this application, we defined x0 = ωd0 (0), and then selected a set of candidate
constants, based on experience with the cases degree d = 3 and d = 4. In particular,
we tried the following input vectors x in our multipair PSLQ computations, where allterms are computed to at least 400-digit precision:x = ωd0 (0), log(2π), ζ 0 (−2), ζ 0 (−4), · · · , ζ 0 (−d + 3)x=ωd0 (0), log(2π), ζ 0 (−2), ζ 0 (−4), · · · , ζ 0 (−d + 2)

for odd dfor even d.

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Experimental
relations producedby multipair PSLQ
0
0
0
0
0
0
0
0
0
0
0

=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ

0

=
ˆ

0

=
ˆ

0

=
ˆ

0

=
ˆ

0

=
ˆ

ω4 (0) + log(2π) − ζ (−2)−ω50 (0) + log(2π) − 2 ζ 0 (−2)
12 ω60 (0) + 12 log(2π) − 35 ζ 0 (−2) − ζ 0 (−4)
4 ω70 (0) − 4 log(2π) + 15 ζ 0 (−2) + ζ 0 (−4)−360 ω80 (0) − 360 log(2π) + 1624 ζ 0 (−2) + 175 ζ 0 (−4) + ζ 0 (−6)
90 ω90 (0) − 90 log(2π) + 469 ζ 0 (−2) + 70 ζ 0 (−4) + ζ 0 (−6)
0−20160 ω10(0) − 20160 log(2π) + 118124 ζ 0 (−2) + 22449 ζ 0 (−4) + 546 ζ 0 (−6) + ζ 0 (−8)
0−4032 ω11 (0) + 4032 log(2π) − 26060 ζ 0 (−2) − 5985 ζ 0 (−4) − 210 ζ 0 (−6) − ζ 0 (−8)
01814400 ω12(0) + 1814400 log(2π) − 12753576 ζ 0 (−2) − 3416930 ζ 0 (−4)
0−157773 ζ (−6) − 1320 ζ 0 (−8) − ζ 0 (−10)
0−302400 ω13(0) + 302400 log(2π) − 2286636 ζ 0 (−2) − 696905 ζ 0 (−4)
0−39963 ζ (−6) − 495 ζ 0 (−8) − ζ 0 (−10)
0239500800 ω14(0) + 239500800 log(2π) − 1931559552 ζ 0 (−2) − 657206836 ζ 0 (−4)
0−44990231 ζ (−6) − 749463 ζ 0 (−8) − 2717 ζ 0 (−10) − ζ 0 (−12)
034214400 ω15(0) − 34214400 log(2π) + 292271616 ζ 0 (−2) + 109425316 ζ 0 (−4)+8691683 ζ 0 (−6) + 183183 ζ 0 (−8) + 1001 ζ 0 (−10) + ζ 0 (−12)
0−43589145600 ω16(0) − 43589145600 log(2π) + 392156797824 ζ 0 (−2) + 159721605680 ζ 0 (−4)
0+14409322928 ζ (−6) + 368411615 ζ 0 (−8) + 2749747 ζ 0 (−10) + 5005 ζ 0 (−12) + ζ 0 (−14)
05448643200 ω17(0) − 5448643200 log(2π) + 51381813456 ζ 0 (−2) + 22556777880 ζ 0 (−4)+2273360089 ζ 0 (−6) + 68396900 ζ 0 (−8) + 654654 ζ 0 (−10) + 1820 ζ 0 (−12) + ζ 0 (−14)
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Solved0 relations
ω4 (0)
ω50 (0)ω60 (0)ω70 (0)ω80 (0)ω90 (0)0ω10(0)0ω11 (0)
0
ω12
(0)
0
ω13
(0)
0
ω14
(0)

=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ
=
ˆ

0
ω15
(0)

=
ˆ

0
ω16
(0)

=
ˆ

0
ω17
(0)

=
ˆ

− log(2π) + ζ 0 (−2)
log(2π) − 2 ζ 0 (−2)
00− log(2π) + 35 ζ12(−2) + ζ (−4)
12
ζ 0 (−4)15 ζ 0 (−2)− 4log(2π) −
4
0
0
0
(−6)
− log(2π) + 203 ζ45(−2) + 35 ζ72(−4) + ζ 360
000
7 ζ (−4)
ζ (−6)
469 ζ (−2)−− 90log(2π) −90
900ζ 0 (−2)ζ 0 (−4)(−6)(−8)− log(2π) + 295315040+ 1069960+ 13 ζ480+ ζ20160
6515 ζ 0 (−2)
95 ζ 0 (−4)
5 ζ 0 (−6)
ζ 0 (−8)log(2π) −−− 96 − 4032
1008
64
0
177133 ζ 0 (−2)
341693 ζ 0 (−4)
ζ 0 (−6)
ζ 0 (−8)(−10)− log(2π) +++ 751386400+ 1115120+ ζ1814400
25200
18144000000190553 ζ (−2)
139381 ζ (−4)
1903 ζ (−6)
11 ζ (−8)ζ (−10)log(2π) −−− 14400 − 6720 − 302400
25200
60480
ζ 0 (−2)
14936519 ζ 0 (−4)
ζ 0 (−6)
ζ 0 (−8)ζ 0 (−10)− log(2π) + 1676701++ 4090021+ 22711+ 2472079005443200
21772800
7257600
21772800ζ 0 (−12)+ 239500800
ζ 0 (−2)ζ 0 (−4)ζ 0 (−6)
ζ 0 (−8)ζ 0 (−10)log(2π) − 634277425− 2486939− 790153− 5551− 913110400
7776003110400
10368000ζ (−12)− 34214400
ζ 0 (−2)ζ 0 (−4)ζ 0 (−6)
ζ 0 (−8)ζ 0 (−10)− log(2π) + 30946717+ 21939781+ 899683+ 515261+ 27473439800
59875202721600
60963840
43545600
0(−12)ζ 0 (−14)+ ζ8709120+ 43589145600
13215487 ζ 0 (−2)
ζ 0 (−4)ζ 0 (−6)
ζ 0 (−8)ζ 0 (−10)log(2π) −− 2065639− 2271089− 4783− 109907200
1401400
4989605443200
38102400(−12)
ζ (−14)− 5448643200− ζ2993760

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Problem: Can we prove these experimental relations?

I This talk is available at

http://www.davidhbailey.com/dhbtalks/dhb-wcnt-2016.pdf.
I A preprint with full details, results and analysis is available at

http://www.davidhbailey.com/dhbpapers/omega-numerics.pdf.

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