Ramanujan-type identities for alternating Hurwitz zeta functions

[2607.03490] Ramanujan-type identities for alternating Hurwitz zeta functions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2607.03490 (math) [Submitted on 3 Jul 2026 ( v1 ), last revised 15 Aug 2026 (this version, v2)] Title: Ramanujan-type identities for alternating Hurwitz zeta functions Authors: Meng Yuan , Su Hu , Min-Soo Kim View a PDF of the paper titled Ramanujan-type identities for alternating Hurwitz zeta functions, by Meng Yuan and 2 other authors View PDF HTML (experimental) Abstract: Around 1910, in an unpublished manuscript, Ramanujan proposed the following identity for $\zeta(2n+1)$: \[ \begin{aligned} \alpha^{-n}\,&\left\{\dfrac{1}{2}\,\zeta(2n + 1) + \sum_{m = 1}^{\infty}\dfrac{m^{-2n - 1}}{e^{2\alpha m} - 1}\right\} \\ &\quad\quad\quad\quad\quad\quad\quad-(-\beta)^{-n}\,\left\{\dfrac{1}{2}\,\zeta(2n + 1) + \sum_{m = 1}^{\infty}\dfrac{m^{-2n - 1}}{e^{2\beta m} - 1}\right\}\\ &=2^{2n}\sum_{k = 0}^{n + 1}\dfrac{(-1)^{k-1}B_{2k}\,B_{2n - 2k + 2}}{(2k)!(2n - 2k + 2)!}\,\alpha^{n - k + 1}\beta^k, \end{aligned} \] where $\alpha$, $\beta$ are positive numbers satisfying $\alpha\beta=\pi^2,n\in\mathbb{N},$ $B_n$ denotes the $n$-th Bernoulli number and $\zeta(z)$ is the Riemann zeta function. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler's famous formula for even zeta values. In this paper, we extend Ramanujan's identity to the alternating Hurwitz zeta function. Then we systematically investigate the properties of the alternating Hurwitz zeta function $\zeta_E(z,x)$, as well as the corresponding Ramanujan-type identities, under different modular symmetry conditions. We also establish infinite series expressions for products of the tangent and hyperbolic tangent functions, and express the Dirichlet lambda function $\lambda(z)$ together with linear combinations of infinite series as convolution sums of special sequences. Furthermore, we define alternating Hurwitz kernels of even and odd orders, and obtain Ramanujan-type identities involving the alternating digamma function $\widetilde{\psi}(x)$ and Euler polynomials $E_n(x)$, as well as transformation formulas between even-order and odd-order alternating Hurwitz kernels. Comments: 48 pages Subjects: Number Theory (math.NT) ; Classical Analysis and ODEs (math.CA) MSC classes: 11B68, 11M06, 33B15, 33B10 Cite as: arXiv:2607.03490 [math.NT] (or arXiv:2607.03490v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2607.03490 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Su Hu [ view email ] [v1] Fri, 3 Jul 2026 16:58:38 UTC (26 KB) [v2] Sat, 15 Aug 2026 08:14:56 UTC (27 KB) Full-text links: Access Paper: View a PDF of the paper titled Ramanujan-type identities for alternating Hurwitz zeta functions, by Meng Yuan and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-07 Change to browse by: math math.CA References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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