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 by n X z Li z m m n n For m Li z log z while for m the function Li z is a higher transcendental func m tion The functions Li z can b e extended ana m 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 m P Li y for all inte tions relating Li x to p 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 c 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 m The main goal of this pap er is to provide an ex extended by the formula ample of ladder relations ie relations among n X z z s Li e s n s z p olylogarithms of p owers of a xed number for s n n 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 m 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 n 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 k k X n nd in the literature k n n k n k n m a natural number and jz j Proposition 1. For k X n n k we have n n e e n X n z z Li e m n m n where denotes B n and for k n n k n n m mo d k for k mo d k m z 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 n 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 m sion of the p olylogarithm function which is dened tion at the mth rung should have the form by X c n n P m m r m X n B r r n P x Re log jxj Li x m m mr r Q r N n c n for some cyclotomic relation However not all relations obtained by pseudoin where Re denotes real or imaginary part dep end m 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 n 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 m 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 i X c n n arithm values of p owers of a single algebraic in S P mi m m n teger are given These recip es which are sp e n 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 mi consider the pseudointegrated quantities Y n c N n i X c n n n S P mi m m n n for some integers c c c for almost all n 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 mi r r 1 2 conjugating the equation that N must b e equal to R However b ecause of the functional equa P nc Since the mo died unilogarithm func n tions of the function P x not all of the com n m 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 dm tor space R of dimension dm dep ending on X n the parity of m Sp ecically since P x is invari m c P n ant under x x for m o dd and antiinvariant for n 28 Experimental Mathematics, Vol.