## Advanced Studies in Pure Mathematics

### An analogue of the Chowla–Selberg formula for several automorphic $L$-functions

Masatoshi Suzuki

#### Abstract

In this paper, we will give a certain formula for the Riemann zeta function that expresses the Riemann zeta function by an infinte series consisting of $K$-Bessel functions. Such an infinite series expression is regarded as an analogue of the Chowla-Selberg formula. Roughly speaking, the Chowla-Selberg formula is the formula that expresses the Epstein zeta-function by an infinite series consisting of $K$-Bessel functions. In addition, we also give certain analogues of the Chawla-Selberg formula for Dirichlet $L$-functions and $L$-functions attached to holomorphic cusp forms. Moreover, we introduce a two variable function which is analogous to the real analytic Eisenstein series and give a certain limit formula for this one. Such a limit formula is regarded as an analogue of Kronecker's limit formula.

#### Article information

Dates
Revised: 5 June 2006
First available in Project Euclid: 27 January 2019

https://projecteuclid.org/ euclid.aspm/1548550911

Digital Object Identifier
doi:10.2969/aspm/04910479

Mathematical Reviews number (MathSciNet)
MR2405616

Zentralblatt MATH identifier
1219.11137

#### Citation

Suzuki, Masatoshi. An analogue of the Chowla–Selberg formula for several automorphic $L$-functions. Probability and Number Theory — Kanazawa 2005, 479--506, Mathematical Society of Japan, Tokyo, Japan, 2007. doi:10.2969/aspm/04910479. https://projecteuclid.org/euclid.aspm/1548550911