On special values of Koshliakov zeta functions
In this paper, we study the Koshliakov zeta function $η_p(s)$, whose theory appears to be more involved than that of its counterpart $ζ_p(s)$, owing to the fact that its defining series is not of Dirichlet type. We derive formulas for $η_p(s)$ 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.