On the Product of Zeta-Functions

Nianliang Wang (), Kalyan Chakraborty and Takako Kuzumaki ()
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Nianliang Wang: School of Applied Mathematics and Computers, Institute of Applied Mathematics, Shangluo University, Shangluo 726000, China
Kalyan Chakraborty: Department of Mathematics, SRM University AP, Amaravati 522240, India
Takako Kuzumaki: Department of Mathematics, Faculty of Engineering, Gifu University, Gifu 501-1193, Japan

Mathematics, 2025, vol. 13, issue 11, 1-15

Abstract: In this paper, we study the Bochner modular relation (Lambert series) for the k th power of the product of two Riemann zeta-functions with difference α , an integer with the Voronoĭ function weight V k . In the case of V 1 ( x ) = e − x , the results reduce to Bochner modular relations, which include the Ramanujan formula, Wigert–Bellman approximate functional equation, and the Ewald expansion. The results abridge analytic number theory and the theory of modular forms in terms of the sum-of-divisor function. We pursue the problem of (approximate) automorphy of the associated Lambert series. The α = 0 case is the divisor function, while the α = 1 case would lead to a proof of automorphy of the Dedekind eta-function à la Ramanujan.

Keywords: Riemann zeta-function; functional equation; modular relation; q -expansion (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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