Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On special values of Koshliakov zeta functions

Yashovardhan Singh Gautam, Rahul Kumar

Abstract

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.

On special values of Koshliakov zeta functions

Abstract

In this paper, we study the Koshliakov zeta function , whose theory appears to be more involved than that of its counterpart , owing to the fact that its defining series is not of Dirichlet type. We derive formulas for at both even and odd values of . In the limiting case , 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 -analogues of the transformation formula for the classical Eisenstein series. Moreover, we introduce two families of -analogues of Ramanujan polynomials and establish functional equations satisfied by them.

Paper Structure

This paper contains 11 sections, 21 theorems, 144 equations.

Table of Contents

  1. Introduction and Motivation
  2. New results for Koshliakov zeta function $\eta_p(s)$ and their consequences
  3. Generalized Ramanujan polynomials
  4. Closed-form evaluations of certain Infinite Series arising from the Transformation Formula for $\eta_p(s)$
  5. $p$-analogue of the Eisenstein series
  6. Preliminaries and basic properties of $p$-analogues of some classical functions
  7. Proofs
  8. Euler's formula for the second Koshliakov zeta function $\eta_p(2m)$
  9. Transformation formula for the second Koshliakov zeta function $\eta_p(s)$ at odd integers
  10. Functional equations for generalized Ramanujan polynomials
  11. Concluding Remarks

Key Result

Theorem 2.1

For $m\in\mathbb{N}$, we have Consequently, we have the following Euler-type formula for $\eta_p(2m)$: where $B_{2m}^{(2,p)}$ are the generalized Bernoulli numbers of the second kind which are defined by the generating function koshliakov where $\sigma_p(t)$ is defined in sigma_p. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (44)

  • Theorem 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Theorem 2.4
  • remark 1
  • Corollary 2.5
  • Corollary 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Corollary 2.9
  • ...and 34 more