A sixteenth-order polylogarithm ladder. (English) Zbl 0763.11049

Cohen, Henri; Lewin, Leonard; Zagier, Don

Exp. Math. 1, No. 1, 25-34 (1992).

The \(m\)th polylogarithm function is defined by \(\text{Li}_ m(z)=\sum^ \infty_{n=1}z^ nn^{-m}\). The modified polylogarithm is defined by \[ P_ m(x)=\text{Re}_ m\left(\sum^ m_{r=0}2^ rB_ r(r!)^{- 1}(\log| x|)^ r\text{Li}_{m-r}(x)\right), \] where \(\text{Re}_ m\) is the real part if \(m\) is odd, and the imaginary part if \(m\) is even. It is desired to find linear relations between \(P\)-values of powers of an algebraic integer.
Starting from \(\prod^ \infty_{n=1}(\alpha^ n-1)^{c_ n}=\zeta\alpha^ N\), where \(c_ n=0\) for almost all \(n\), and \(\zeta\) is a root of unity, one obtains \(\sum^ \infty_{n=1}c_ nP_ 1(\alpha^ n)=0\). Now \(P_ 1(\alpha^ n)\) is replaced by \(n^{-(m- 1)}P_ m(\alpha^ n)\) to give \(\sum^ \infty_{n=1}c_ nn^{-(m- 1)}P_ m(\alpha^ n)=0\), but not all these relations are true. To establish conjecturally which ones are true the functions \(P\) are evaluated to a large number of decimal places (up to 305). At each increase of \(m\) some relations have to be discarded: the authors start with enough relations to reach \(m=16\). One of the conjectured relations is given explicitly; it involves coefficients with up to 71 digits.

Reviewer: H.J.Godwin (Egham)

MSC:

11Y99	Computational number theory
33E99	Other special functions

Keywords:

numerical computation of polylogarithms; LLL algorithm; ladder method; linear relations between \(P\)-values of powers of an algebraic integer; modified polylogarithm