On special values of Koshliakov zeta functions
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866915917848379392 |
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| author | Gautam, Yashovardhan Singh Kumar, Rahul |
| author_facet | Gautam, Yashovardhan Singh Kumar, Rahul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04675 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On special values of Koshliakov zeta functions Gautam, Yashovardhan Singh Kumar, Rahul Number Theory Classical Analysis and ODEs 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. |
| title | On special values of Koshliakov zeta functions |
| topic | Number Theory Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2604.04675 |