## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Probability and Number Theory — Kanazawa 2005, S. Akiyama, K. Matsumoto, L. Murata and H. Sugita, eds. (Tokyo: Mathematical Society of Japan, 2007), 479 - 506

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

#### 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**

Received: 11 January 2006

Revised: 5 June 2006

First available in Project Euclid:
27 January 2019

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/04910479

**Mathematical Reviews number (MathSciNet)**

MR2405616

**Zentralblatt MATH identifier**

1219.11137

**Subjects**

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$

Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11M20: Real zeros of $L(s, \chi)$; results on $L(1, \chi)$ 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

#### 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