A Sixteenth-Order Polylogarithm Ladder Henri Cohen, Leonard Lewin and Don Zagier

A Sixteenth-order Polylogarithm Ladder Henri Cohen, Leonard Lewin and Don Zagier

CONTENTS Using the LLL algorithm and the second author’s ‘‘ladder’’ 1. Introduction method, we find (conjectural) Z-linear relations among poly- 2. Numerical Computation of Polylogarithms logarithms of order up to 16 evaluated at powers of a single 3. Ladders algebraic number. These relations are in accordance with 4. Climbing the Ladder theoretical predictions and are valid to an accuracy of 300 5. Numerical Results decimal digits, but we cannot prove them rigorously. Appendix: A (Conjectural) Hexadecalogarithm Relation References

1. INTRODUCTION

Let m b e a p ositive integer The mth p olyloga

rithm function is dened for complex z with jz j

Li z

For m Li z log z while for m

the function Li z is a higher transcendental func

tion The functions Li z can b e extended ana

lytically to the whole complex plane if one makes

a cut for example the real line from to innity

corresp onding for m to the principal branch

of the logarithm

The function Li z satises the functional equa

tion

Li xy Li x Li y

which implies that any linear combination with in

tegral co ecients of Li values of elements of a

number eld will again b e an Li value of an el

ement of this number eld The corresp onding

statement for higher p olylogarithms is not true

However the Li satisfy trivial functional equa

Li y for all inte tions relating Li x to

m m

y x

gers p and for small orders m they also sat

isfy more complicated nontrivial functional equa

tions The present record due to H Gangl gives

nontrivial functional equations for m In

addition these functional equations give relations

among p olylogarithm values of elements of a given

A K Peters

p er page

26 Experimental Mathematics, Vol. 1 (1992), No. 1

number eld It is conjectured that al l relations The pro of by induction is straightforward We ob

among p olylogarithms of algebraic arguments can serve as an amusing remark that the p olyloga

b e obtained by sp ecializing functional equations rithm of complex order s dened for jz j like

but this is not known for any m Li but with m replaced by s can b e analytically

The main goal of this pap er is to provide an ex extended by the formula

ample of ladder relations ie relations among

z s

Li e s n s z

p olylogarithms of p owers of a xed number for

the number dened as the unique ro ot

outside the unit circle of the selfrecipro cal p olyno

for all s N the limit of the righthand side as

mial

s m N is the expression o ccurring in the

prop osition

X X X X X X X X

i z

If x e C and we write x e the condi

tion jz j is equivalent to

Dilogarithm relations for this number were rst in

p p

vestigated by G Ray and are given in Chapter of

exp exp

Lewin The ladder relations we will nd go

Since we may assume that j j this region con

up to order conrming the general theory as

p p

e note that tains the annulus e

describ ed in Zagier a and recalled in Section

p p

This is quite p ossibly the highest order o ccur

e and e So to compute

ring for any algebraic number a prop erty related

Li x we use the p ower series if jxj is small say

to the wellknown conjecture of Lehmer that

jxj the formula of prop osition if jxj is near

has the smallest p ossible Mahler measure for an

say jxj and the functional equation re

algebraic integer that is not a ro ot of unity the

lating Li x to Li x if jxj is large say jxj

m m

Mahler measure of an algebraic integer is equal to

in fact as we will see we will never have to use

the pro duct of the mo duli of the numbers con

this

jugates that are greater than hence here equal

There is still one more p oint to treat We will

to The computation of the ladder relations

need p olylogarithm values to several hundred dec

requires b oth highaccuracy computation of p oly

imals hence we also need the values of k for

logarithms we needed more than decimal dig

integers k to such a high accuracy When k

its and careful use of linear algebra and lattice

or k and even k is given explicitly in terms

reduction techniques for fairly large matrices with

of Bernoulli numbers B We must also compute

nonexact entries so b oth of these issues will have

k for k o dd k We could use the stan

to b e addressed in the pap er

dard EulerMcLaurin metho d for doing this but

there is a simpler and much faster metho d coming

the theory of mo dular forms more precisely

2. NUMERICAL COMPUTATION OF from

integrating k times the Fourier expansion of the

POLYLOGARITHMS by

holomorphic Eisenstein series of weight k these

To achieve our goal it is necessary to compute

formulas were known to Ramanujan

p olylogarithms to a reasonably high accuracy To

For k odd k we have

this we use the following very simple and pretty

do Proposition 2.

formula for the p olylogarithm which we could not

nd in the literature

k n

m a natural number and jz j

Proposition 1. For

n n

we have

n n

e e

Li e m n

where denotes B n and for k

n n

n m

mo d k for k mo d

The ab ove prop ositions give us rapidly conver

log z

m m

gent metho ds for computing p olylogarithms in the

Cohen, Lewin and Zagier: A Sixteenth-order Polylogarithm Ladder 27

whole complex plane and in particular near or on So these are our unilogarithm relations the rst

the unit circle Of course Prop osition is not re rung of the ladder The main statement of the

ally necessary since the values of k for k up to theory is that all relations among mth p olyloga

could b e computed just once by a slower metho d rithms are obtained by m fold pseudointegra

and then stored tion that is the formal replacement of P by

m n

We will actually b e working with a mo died ver

n P In other words any ladder rela

sion of the p olylogarithm function which is dened

tion at the mth rung should have the form

P x Re log jxj Li x

m m mr

N n c

for some cyclotomic relation

However not all relations obtained by pseudoin

where Re denotes real or imaginary part dep end

tegration will hold The rest of the picture is as

ing on whether m is o dd or even and Li has

follows First observe that the set of cyclotomic

Zagier a In contrast b een set equal to

equations forms a group under multiplication so

with the original p olylogarithms these mo died

the set of p ossible collections of exp onents fc g

functions satisfy clean functional equations ie

forms a group under addition obviously the same

equations involving only p olylogarithms of a given

is true for the linear relations among the p olylog

order without lowerorder correction terms In

arithms of any given order Supp ose that we have

particular we have P x P x

m m

found I multiplicatively indep endent cyclotomic

relations Then starting with m supp ose in

3. LADDERS

ductively that we have I m linearly indep en

dent equations at order m say

In Ab ouzahra and Lewin and Lewin

recip es to obtain linear relations among p olylog

arithm values of p owers of a single algebraic in

S P

mi m

teger are given These recip es which are sp e

cial cases of the theory describing linear relations

for i I m We supp ose further that the

among p olylogarithm values of arbitrary algebraic

same relation remains true if is replaced by any

arguments as explained in Zagier a b

of its conjugates In practice and according

are as follows

to conjectures this is automatically true but in

Assume that one knows for some algebraic num

any case we need it for the inductive setup We

b er a certain number of relations of the form

write this symbolically as S We then

consider the pseudointegrated quantities

n c N

S P

mi m

for some integers c c c for almost all

n and N and some ro ot of unity Such rela

There are r r p ossible embeddings of Q

tions are called cyclotomic relations For the sp e

into the complex numbers where r and r denote

cial number dened in the introduction since is

as usual the number of real and complex conju

real and greater than we must have more

gates of so for each i the collection of num

over since is a conjugate of we deduce by

b ers fS g can b e considered as a vector in

r r

1 2

conjugating the equation that N must b e equal to

R However b ecause of the functional equa

nc Since the mo died unilogarithm func

tions of the function P x not all of the com

tion P is given by P x log j xj log jxj

p onents of this vector are indep endent so we can

the cyclotomic equation can b e rewritten

think of it as a vector in a smaller dimensional vec

tor space R of dimension dm dep ending on

the parity of m Sp ecically since P x is invari

c P

ant under x x for m o dd and antiinvariant for

28 Experimental Mathematics, Vol. 1 (1992), No. 1

The rst step is to get the ladder started by m even we need only half of the complex conju

nding cyclotomic relations for the number as gates and can ignore the real conjugates when m

explained at the b eginning of this section We do is even reducing the number of comp onents from

this as follows First notice that the p olynomial r r to r r or r in our sp ecial case and

X factors as a pro duct of cyclotomic p olyno resp ectively Moreover for like ours satisfying a

mials X over all divisors k of n so that nding selfrecipro cal equation the functional equation

multiplicative relations among and the numbers

P x P x

is the same as nding multiplicative rela

m m

tions among and the numbers Any such

implies that we also need only half of the real em

relation will imply a relation among the norms so

b eddings when m is o dd ie dm is r in our

we rst compute the norm of for many k

case when m is even and r r in our

We nd that this norm is for values of k less

case when m is o dd Now the theory says that

than namely

the vectors obtained by pseudointegrating the valid

m th order relations will lie in a lattice in

R or in other words that any dm of

them will satisfy a linear relation over Z rather

than merely over R This assertion given con

jecturally in Zagier a is actually now a the

orem to app ear in forthcoming work by Beilin

son and Deligne In particular at least I m

But even when the norm of is not a unit

I m dm of our I m vectors in R will

we can hop e to multiplicatively combine several

b e Changing bases if necessary we can supp ose

to obtain a unit and this indeed happ ens

that these are simply the relations S for

giving us the nine further units for

k k

1 2

i I m and this puts us back in the same

k k

situation as b efore and lets us continue the induc

tive pro cess Summarizing in climbing the lad

der we lose at most dm relations at each step

Together with itself this gives us multiplica

and hence starting with the I cyclotomic rela

tively indep endent units and hence since the unit

tions can mount the ladder as long as the quantity

rank of K Q is equal to r r lin

I m I d dm remains p ositive

early indep endent cyclotomic relations The ve

Note that although the theory tells us that such re

units and for k form a generat

lations exist it do es not give any b ound on the size

ing system of indep endent units

of the co ecients As a consequence although the

Using Bakertype metho ds it is p ossible to give

relations which we will give are correct to several

an eective upp er b ound on the index of the cy

hundred decimal places and although the number

clotomic p olynomials which have to b e examined

and type of relations we nd agrees with the num

However as often with this type of b ound it do es

b er which we know must exist the sp ecic relations

not seem p ossible to decrease it to a computation

that we give are not rigorously proved

ally feasible value Hence although it is p ossi

A striking example of a ladder of the sort just

ble that some cyclotomic relation has escap ed our

describ ed is given in Ab ouzahra et al and

search this seems quite unlikely

Zagier a where one takes for a ro ot of

X In this case one can climb the

X 4. CLIMBING THE LADDER

ladder up to level and at level there remain

pseudointegrated relations which are just su By the construction sketched ab ove from our ini

cient to give a relation b etween nonalogarithms if tial I cyclotomic relations we must ob

one adds P P to the list of ad tain I dilogarithm relations I

missible p owers of In the present pap er we trilogarithm relations decreasing alternatively by

study the sp ecial Salem number mentioned in the or and nally we must obtain I hex

introduction an even more sp ectacular example adecalogarithm relations We would need six such

Cohen, Lewin and Zagier: A Sixteenth-order Polylogarithm Ladder 29

Because M has integer entries and we are only relations to b e able to climb to the th level To

concerned with the pro duct of this matrix with K obtain these relations is not a completely trivial

we may as well rst compute task and in this section we explain the metho d

that has b een used to obtain them All the com

L M K

putations were done using the GPPARI calculator

developed by C Batut D Bernardi H Cohen and

and then try to approximate it by a matrix with ra

M Olivier

tional entries If we are given sucient accuracy

There are exactly dierent exp onents of

we can simply use the continued fraction expan

which o ccur in the cyclotomic relations Hence we

sion of the co ecients and hop e to nd a reason

will represent the cyclotomic relations as a

ably simple denominator In the present case the

matrix L with integral entries each column

co ecients are apparently all integers so we have

representing a relation This matrix is of course

nothing more to do than rounding but in higher

not unique and we can try to get the co ecients

levels we will have to nd a common denominator

as small as p ossible by using the LLL algorithm

and dene L by multiplying L by this denom

Lenstra et al We can give a rough measure

inator and rounding Even here if we had not

of the size of the co ecients by computing the L

LLLreduced the matrix L we would have found

norm of L In the present case the minimal value

the rather small denominator

found and probably the absolute minimum is

If L is our rounded matrix we can say with

Now we pseudointegrate these relations which

condence esp ecially if we work with decimal

simply means that we divide each of the rows

digits which we will need for higher levels that we

j by its corresp onding exp onent from to

have found the correct relations at the diloga

which we denote by ej Let M b e the resulting

rithm level This is however still not entirely sat

matrix We form the matrix V v with

isfactory since we may not have a Zbasis of the

lattice of relations but only a Qbasis

v P where the are four nonreal con

ij i

To check whether we have a Zbasis we use the

jugates of one for each complex conjugate pair

following easily proved lemma

and then form the matrix V M of tuples

of dilogarithm values for the pseudointegrated

M be an m n matrix with integer

Lemma. Let

relations This matrix turns out to have maximal

entries and rank n Then the lattice spanned by the

rank so we pick four indep endent columns ac

columns of M is equal to the intersection with Z

tually the rst four and then form the

of the Qvector space spanned by the columns of M

matrix

if and only if the GCD of al l n n subdeterminants

K kerV M

is equal to

whose columns are the relations obtained by We will say that a matrix satisfying the condi

writing each of the remaining last columns tion of the preceding lemma is primitive To re

of V M as a linear combination of the chosen place a matrix M by a primitive one whose columns

four Notice that this pro cedure is numerically generate the same Qvector space we can for in

welldened and stable even though the entries of stance pro ceed as follows We extract at random

our matrices are only approximate b ecause the two n n determinants and let d b e their GCD

invertibility of the matrix dep ends on the Then for every prime p dividing d we make M

nonvanishing of a number and this is something primitive at p by elementary linear algebra mo d

which can b e checked numerically The theory ulo p Note that for this we do not need to factor d

describ ed in Section tells us that K should in completely since once the small prime factors have

fact have rational co ecients and it is then easy b een removed we can treat the remaining part as

to see that a p ossible choice for the matrix L a prime and the only thing which can go wrong is

of relations for dilogarithms would then b e L that we obtain a nontrivial factor of this prime

M K in which case we try again with that factor and its

The rst problem that faces us is that K has cofactor Alternatively we could apply Hermite

only b een calculated approximately so we have to reduction to M after which it is obvious how to

somehow recognize its entries as rational numbers make it primitive

30 Experimental Mathematics, Vol. 1 (1992), No. 1

relations b etween real numbers However we did Applying this pro cedure to L we now obtain

not succeed with that either The idea which a Zbasis of the space of relations not just a Q

nally worked is very simple Before computing the basis As b efore we terminate that matrix by LLL

kernel do an LLLreduction of the matrix M It reducing In our case if we start from the LLL

is not unreasonable to exp ect that this will lower reduced matrix L the GCD d of sub determinants

not only the size of the co ecients of M but is already equal to so the reduction step is unnec

also that of the denominators This is in fact essary On the other hand if we had started from a

strikingly true see next section and enabled us nonLLLreduced matrix L giving a denominator

to climb to the top of the ladder ie m in we would have had to eliminate ie and

a few more minutes and thus to obtain the desired from the GCD of the sub determinants

four hexadecalogarithm relations b etween p owers Since we LLLreduce each matrix L as we pro

of ceed the relations that we obtain are essentially

optimal Finding a kernel is no problem since it

inverting a or matrix followed

involves Remarks

by a matrix multiplication Similarly the pro cess

It would b e of considerable interest to develop

of making a matrix primitive is fairly straightfor

an algorithm which can compute directly an LLL

ward The crucial problem which arises now is that

reduced Zbasis of relations b etween linearly de

of explosion of denominators Since the kernel is

p endent vectors v v without computing rst

computed in an arbitrary way at rst and only

a Qbasis of Qv then transforming this into a

corrected afterwards it is not surprising that we

Zbasis and nally LLLreducing The algorithm

have little control on the denominators which o c

should b e able to deal with exact or imprecise en

cur In fact working still with decimals of

tries ie we should not have to rst nd an R

accuracy we are able to go not much further than

basis and then recognize the real entries as ra

level or So we must still improve our strat

tional numbers One way to do this is to apply

egy

LLLreduction to the lattice Z equipp ed with the

A rst improvement is as follows It is clear that

quadratic form

the pseudointegrated matrices M after clearing

Qa a ka v a v k

n n n

denominators may not b e primitive in the ab ove

a a

sense In fact the GCD of maximal sub determi

nants will b e divisible by all the primes dividing

for a a Z with a suitably small then

the exp onents of the cyclotomic relations In our

the very short vectors corresp ond to the kernel

case this corresp onds exactly to all the primes

and automatically give an LLLreduced basis of it

up to and including We can make the ma

When the Euclidean space in which the v lie is

trices M primitive with resp ect to all of these

onedimensional this is the standard way of nd

primes and exp eriment shows that no other primes

ing integer relations among real numbers This

need to b e eliminated With this improvement we

algorithm can b e made to work but requires care

were able to reach level However once again

ful choice of

we were caught up by the explosion of the size

Initially instead of using the conjugates of

of the denominators ie after computing L

we simply lo oked using LLL for relations among

M kerV M we were unable to recognize

the unmo died p olylogarithm functions Li eval

the common denominator of the co ecients of L

uated at p owers of the real number However

At that p oint several ideas were tried First

the size of the denominators and of the co ecients

we could simply have increased the precision of the

which o ccur increases even more rapidly than in

p olylogarithm values but a rough estimate showed

the metho d we nally used and we were unable to

that we probably would have exceeded reasonable

climb b eyond level

computer storage and time limitations Second

must have a common since the co ecients of L

5. NUMERICAL RESULTS

denominator it might b e p ossible to use the LLL

algorithm to nd it by using a version which is es Let us summarize the pro cedure describ ed ab ove

sentially dual to the version used in nding linear Let L b e the LLLreduced matrix of cy

Cohen, Lewin and Zagier: A Sixteenth-order Polylogarithm Ladder 31

clotomic relations Consider some integer m

m d l

m m m

and assume inductively that the matrix L has

b een computed Let M b e the matrix obtained

after p erforming successively the following steps

a pseudointegrate the matrix L

b reduce the columns to integer entries having

GCD equal to

c eliminate the primes up to

d LLLreduce

Let V b e the dm dm or matrix

of p olylogarithm values of the p owers of the rele

M kerV M vant conjugates of Set L

m m m

where the kernel is computed by simple matrix in

version Using the continued fraction expansion of

determine a plausible de a few co ecients of L

nominator d for L and let L b e the matrix

m m

obtained from L by the following steps

a Multiply L by d and then round each co ef

cient to the nearest integer let b e the sum

of the absolute values of the distances b etween

Salem number ladder data

co ecient and its rounded value

each TABLE 1.

b Eliminate d from the resulting matrix In

we suer a loss of precision of ab out the same or

practice eliminate individually the small prime

der as d l In particular if we had worked with

m m

factors of d and globally the remaining cofac

only decimals instead of we would not

tor even if it is not prime

have b een able to obtain the hexadecalogarithm

c LLLreduce

relations By using the metho d sketched earlier in

In Table we give the results obtained by work Remark preceding this section we could have

ing with decimals of accuracy Note that since avoided the denominator explosion altogether but

there is some freedom in the LLLreduction pro apparently the same precision ab out digits

cedure another implementation may give slightly would still have b een needed to obtain the highest

dierent results although of course the lattices of order relation

p olylogarithm relations will b e the same We give The third remark is that the relations have b een

successively m d and l the L norm of the LLLreduced hence the L norms given in the last

m m m

LLLreduced matrix L column are certainly close to b est p ossible up to

A few remarks are in order concerning this ta maybe a variation of or in the exp onent of

ble First thanks to the combination of all the In particular the four relations that we obtain

tricks mentioned ab ove the explosion of denomi at level have digit co ecients and the LLL

nators has b een completely avoided for small levels b ounds tell us that no such relation exists whose

and in fact the denominator is equal to for all co ecients have at most digits Just for fun

even levels up to On the other hand when we have given one of these relations in the Ap

the level is or ab ove the denominators b ecome p endix

very large We b elieve that the reason for this is The last remark is the following At even or

that the matrices M get narrower as m grows ders m we have worked with four nonreal con

for example M has only eight columns hence jugates of since this is enough to obtain the

the LLLreduction of these matrices which saved relations and also b ecause the mo died p olylog

us in the end is unfortunately not of much help A arithm function P vanishes at real arguments

related observation is that up to level the pre However the theory tells us that for each rela

cision of the results stays excellent but thereafter tion c P the corresp onding com

n m

32 Experimental Mathematics, Vol. 1 (1992), No. 1

cyclotomic elds one exp ects at most the same bination c L of the mo died real p olylog

n m

denominator as for Q ie at most D if we write arithm function

m m as N D with N and D

m m m m

log jxj

coprime integers For instance all known rela

L x Li x

m mr

tions among values of the values of the function

L Rogers dilogarithm function at arguments

log jxj

log j xj

in real quadratic elds other than Q give in

tegral multiples of The question was

for m even and x should give a rational

whether the right general conjecture should b e that

multiple of This is indeed what we nd More

p olylogarithm relations give simple rational multi

over the calculation lets us answer a question that

ples of or simple rational multiples of m

had arisen in earlier p olylogarithm investigations

ie whether for generic F they are integral mul

but could not previously b e answered b ecause the

m m

tiples of N D or merely of D This

m m m

orders of the p olylogarithms o ccurring were to o

could not b e checked b efore b ecause no relations

small In all known relations among the values

b eyond order were known and the rst m with

of the real p olylogarithm function L at real ar

N is N It turned out that for

guments the combination of p olylogarithms which

m and m the rational multiples o ccur

o ccur are not merely rational multiples of but

ring did not have numerators divisible by N For

are rational multiples with a small denominator

instance the combination of mo died hexadecalog

Conjecturally a prime p can o ccur in the denomi

arithms given in the App endix is an integer mul

nator only if some extension of F of degree m con

tiple of D but not of

tains a pth ro ot of unity in particular for generic

elds not having esp ecially large intersections with

APPENDIX: A (CONJECTURAL) HEXADECALOGARITHM RELATION

As promised we give one of the four relations among the hexadecalogarithm values of p owers of We give

the co ecients in tabular form each line containing an integer n b etween and and a co ecient c of

c P vanishes for each conjugate of As mentioned at most digits such that the sum

in Section these relations are not rigorously proved even though we know that relations of this form

exist The value of the sum c L which should b e a multiple of turns out to b eat least

to digit accuracyequal to

n c

Cohen, Lewin and Zagier: A Sixteenth-order Polylogarithm Ladder 33

34 Experimental Mathematics, Vol. 1 (1992), No. 1

REFERENCES

Lewin L Lewin The inner structure of the

Ab ouzahra and Lewin M Ab ouzahra and L

dilogarithm in algebraic elds J Number Th

Lewin The p olylogarithm in algebraic number

elds J Number Th

Lewin L Lewin ed The Structural Properties

Ab ouzahra et al M Ab ouzahra L Lewin and

of Polylogarithms Amer Math So c Monographs

H Xiao Polylogarithms in the eld of omega a ro ot

American Mathematical So ciety Providence RI

of a given cubic functional equations and ladders

Aeq Math

Zagier a D Zagier Polylogarithms Dedekind

Lehmer D H Lehmer Factorization of certain

zeta functions and the algebraic K theory of elds

cyclotomic functions Ann Math

pp in Arithmetic Algebraic Geometry ed

ited by G vd Geer F Oort J Steenbrink Prog

Lenstra et al A Lenstra H Lenstra and

in Math Birkhauser Boston

L Lovasz Factoring p olynomials with rational

Zagier b D Zagier Sp ecial values and func

co ecients Math Ann

tional values of p olylogarithms App endix in Lewin

Lewin L Lewin Polylogarithms and Associated

Functions NorthHolland New York

Henri Cohen Centre de Recherche en Mathematiques de Bordeaux Universite Bordeaux I Talence France

cohenaliothgrecoprogfr

Leonard Lewin Department of Electrical and Computer Engineering University of Colorado at Boulder Campus

Box Boulder CO

Don Zagier MaxPlanckInstitut f ur Mathematik GottfriedClarenStr Bonn Germany

Received September corrected in pro ofs February