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arXiv · 2112.09322

A new Ramanujan-type identity for $L(2k+1,\chi_1)$

Shashi Chourasiya·Md Kashif Jamal·Bibekananda Maji
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Abstract

One of the celebrated formulas of Ramanujan is about odd zeta values, which has been studied by many mathematicians over the years. A notable extension was given by Grosswald in 1972. Following Ramanujan's idea, we rediscovered a Ramanujan-type identity for $\zeta(2k+1)$ that was first established by Malurkar and later by Berndt using different techniques. In the current paper, we extend the aforementioned identity of Malurkar and Berndt to derive a new Ramanujan-type identity for $L(2k+1, \chi_1)$, where $\chi_1$ is the principal character modulo prime $p$. In the process, we encounter a new family of Ramanujan-type polynomials and we notice that a particular case of these polynomials has been studied by Lal\'{i}n and Rogers in 2013. Furthermore, we establish a character analogue of Grosswald's identity and a few more interesting results inspired from the work of Gun, Murty and Rath.

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Shashi Chourasiya, Md Kashif Jamal, Bibekananda Maji. 2021-12-17. A new Ramanujan-type identity for $L(2k+1,\chi_1)$. https://arxiv.org/abs/2112.09322

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