Advanced Studies in Pure Mathematics

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

Masatoshi Suzuki

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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

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

Received: 11 January 2006
Revised: 5 June 2006
First available in Project Euclid: 27 January 2019

Permanent link to this document euclid.aspm/1548550911

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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}


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.

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