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A new Ramanujan-type identity for \(L(2k+1, \chi _1)\)

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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. Furthermore, we establish a character analogue of Grosswald’s identity.

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Acknowledgements

We sincerely thank the anonymous referee for carefully reading our manuscript and providing various corrections. The authors would also like to thank Prof. Bruce C. Berndt and Prof. Atul Dixit for giving valuable suggestions. We are also thankful to the Computational Number Theory (CNT) Lab, IIT Indore for providing conductive research environment.

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  1. Department of Mathematics, Indian Institute of Technology Indore, Simrol, Indore, Madhya Pradesh, 453552, India

    Shashi Chourasiya, Md Kashif Jamal & Bibekananda Maji

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  1. Shashi Chourasiya
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  2. Md Kashif Jamal
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  3. Bibekananda Maji
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Correspondence to Bibekananda Maji.

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Dedicated to Srinivasa Ramanujan on the 134th anniversary of his birth

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The third author thanks the SERB for the Start-Up Research Grant SRG/2020/000144.

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Chourasiya, S., Jamal, M.K. & Maji, B. A new Ramanujan-type identity for \(L(2k+1, \chi _1)\). Ramanujan J 60, 729–750 (2023). https://doi.org/10.1007/s11139-022-00661-6

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