Item

Abstract

The mth polylogarithm function is defined by \textLi\sb m(z)=\sum\sp ∞\sbn=1z\sp nn\sp-m. The modified polylogarithm is defined by P\sb m(x)=\textRe\sb m≤ft(\sum\sp m\sbr=02\sp rB\sb r(r!)\sp- 1(\log\vert x\vert)\sp r\textLi\sbm-r(x)\right), where \textRe\sb 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.\par Starting from \prod\sp ∞\sbn=1(α\sp n-1)\spc\sb n=ζα\sp N, where c\sb n=0 for almost all n, and ζ is a root of unity, one obtains \sum\sp ∞\sbn=1c\sb nP\sb 1(α\sp n)=0. Now P\sb 1(α\sp n) is replaced by n\sp-(m- 1)P\sb m(α\sp n) to give \sum\sp ∞\sbn=1c\sb nn\sp-(m- 1)P\sb m(α\sp 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.