POLYLOGARITHM APPROACHES TO RIEMANN ZETA ZEROES

We use visible point vector identities to examine polylogarithms in the neighbourhood of the Riemann zeta function zeroes. New formulas limiting to the trivial zeroes and to the critical line on the zeta function are given. Similar results arise from Euler-Zagier sums given in Bailey, Borwein and Crandall (4) providing new infinite product identities. 1. Polylogarithms near trivial zeroes of the Riemann zeta function In this note we examine the limit as we approach the zeroes of the Riemann zeta function from the polylogarithm function. Although we will not hope to prove any- thing as profound as the Riemann Hypothesis itself, we can examine the behaviour of the polylogarithm in the vicinity of the trivial and non-trivial zeroes of the Rie- mann zeta function, and along the way prove some new infinite product identities. The two examples of Visible Point Vector identities cited in WolframMathWorld (see Weisstein (21)) are cases of the authors identities, which include the Theorem 1.1. (Campbell (9)) If |x|,|y|,|z| < 1 and s,t,u, are complex numbers and Lis(z) = P 1=1 z k k s ,