Laurent coefficients of Zagier-type zeta function {à} la Ishibashi and arithmetic aspects of extended Ramanujan period function
Abstract
One of the remarkable contributions of Don Zagier was the Kronecker limit formula for a real quadratic field, where he connects the double series 𝒵(s,w,w′) to the Dedekind zeta function associated to a real quadratic field. Later, Ishibashi determined all the Laurent coefficients of 𝒵(s,w,w′) at s=1. Recently, Choie and kumar have studied the analytic behaviour of the analogous double series 𝒵̃(s,w,w′). In this article, we derive all the Laurent coefficients of 𝒵̃(s,w,w′), akin to Ishibashi. These Laurent coefficients involve an interesting function Fk0(x), which was earlier studied by Dixit et. al. (Ramanujan for k=1), where they obtained a beautiful symmetric relation for Fk0(x). We establish both the two term and the three term functional equation of Fk0(x), derive the action of the period-like Hecke operator on Fk0(x) and connect an important integral with Fk0(x).
Related
- Ramanujan Conjecture and Riemann Hypothesis for the unitary groups over function fields
- Connection Formulae for a Generalised Ramanujan Entire Function
- A number field extension of Ramanujan's identity and its integral representation via the generalized Riemann Ξ-function
- Ramanujan-type congruences for the smallest parts function
- Strongly Consistent Estimation of the Extended ℓ1-Sum of ϕ-Mixing Coefficients from a Single Trajectory
BibTeX
@misc{banerjee2026laurent,
title = {{Laurent coefficients of Zagier-type zeta function à la Ishibashi and arithmetic aspects of extended Ramanujan period function}},
author = {Soumyarup Banerjee and Riya Mandal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15052},
eprint = {2609.15052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15052},
}
RIS
TY - UNPB
TI - Laurent coefficients of Zagier-type zeta function {à} la Ishibashi and arithmetic aspects of extended Ramanujan period function
AU - Soumyarup Banerjee
AU - Riya Mandal
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15052
UR - https://arxiv.org/abs/2609.15052
AB - One of the remarkable contributions of Don Zagier was the Kronecker limit formula for a real quadratic field, where he connects the double series 𝒵(s,w,w^′) to the Dedekind zeta function associated to a real quadratic field. Later, Ishibashi determined all the Laurent coefficients of 𝒵(s,w,w^′) at s=1. Recently, Choie and kumar have studied the analytic behaviour of the analogous double series 𝒵̃(s,w,w^′). In this article, we derive all the Laurent coefficients of 𝒵̃(s,w,w^′), akin to Ishibashi. These Laurent coefficients involve an interesting function F_k^0(x), which was earlier studied by Dixit et. al. (Ramanujan for k=1), where they obtained a beautiful symmetric relation for F_k^0(x). We establish both the two term and the three term functional equation of F_k^0(x), derive the action of the period-like Hecke operator on F_k^0(x) and connect an important integral with F_k^0(x).
ER -