On special values of Koshliakov zeta functions

TL;DR

This paper investigates the properties of the Koshliakov zeta function, deriving formulas at various values, connecting to classical results, and introducing p-analogues of Ramanujan polynomials and Eisenstein series transformations.

Contribution

It provides new formulas for $ ext{eta}_p(s)$, explores their limits, and introduces p-analogues of Ramanujan polynomials and Eisenstein series transformations.

Findings

Formulas for $ ext{eta}_p(s)$ at even and odd $s$

Limit $p o \infty$ recovers Euler and Ramanujan formulas

Introduces p-analogues of Ramanujan polynomials and Eisenstein series

Abstract

In this paper, we study the Koshliakov zeta function $\eta_p(s)$, whose theory appears to be more involved than that of its counterpart $\zeta_p(s)$, owing to the fact that its defining series is not of Dirichlet type. We derive formulas for $\eta_p(s)$ at both even and odd values of $s$. In the limiting case $p\to\infty$, our results yield the celebrated formulas of Euler and Ramanujan for the Riemann zeta function. Moreover, our results lead to several consequences concerning closed-form expressions for Lambert series and their arithmetic properties, recovering results due to Berndt, Cauchy, Ramanujan, and others. We also propose $p$-analogues of the transformation formula for the classical Eisenstein series. Moreover, we introduce two families of $p$-analogues of Ramanujan polynomials and establish functional equations satisfied by them.