On the Product of Zeta-Functions

Nianliang Wang (), Kalyan Chakraborty and Takako Kuzumaki ()
Additional contact information
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
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/11/1900/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/11/1900/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:11:p:1900-:d:1672939

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().