Ramaswami Type translation formulae for the polylogarithm functions
Pawan Singh Mehta, Biswajyoti Saha
TL;DR
The paper extends Ramaswami and Apostol translation formulae from the Riemann zeta function to polylogarithms $\mathrm{Li}_z(s)$ with $z$ a root of unity, establishing a general translation identity for $\Re(s)>1$ and a class of corollaries. Central to the method is a Lerch zeta function-based framework that yields a key expansion $\phi(s,a+1,z)=\frac{1}{z}\sum_{m\ge0}(-a)^m P_m(s)\mathrm{Li}_z(s+m)$, which, after summing over lattice points and employing Euler polynomials, produces $\mathrm{Li}_z(s)(1-k^{\delta_q-s})=\sum_{m\ge1}\frac{P_m(s)\mathrm{Li}_z(s+m)}{k^{s+m}}\sum_{h=1}^{k-1} z^{-h}h^m$ and its variants. Applications include new recurrence relations for even Bernoulli numbers, and detailed congruence analyses for tangent numbers, including parity-based last-digit results and nondivisibility by $3$, as well as $\zeta$-series representations for rationals and special values. The results enhance analytic continuation techniques for polylogarithms and provide tools for arithmetic insights into Bernoulli and tangent numbers.
Abstract
In 1934, Ramaswami proved a number of curious translation formulae satisfied by the Riemann zeta function. Such translation formulae, in turn give the meromorphic extension of the Riemann zeta function. In 1954, Apostol extended those identities to establish a family of such similar translation formulae. In this article, we establish many such Ramaswami and Apostol type translation formulae for the Dirichlet series defining the polylogarithm functions. This extended set up has many interesting applications, for example, it allows us to also find some (seemingly new) recurrence relations between the Bernoulli numbers, and use them to deduce some congruence properties of the tangent numbers.