Open Access
November 2010 The discrete mean square of Dirichlet $L$-function at integral arguments
Guodong Liu, Nianliang Wang, Xiaohan Wang
Proc. Japan Acad. Ser. A Math. Sci. 86(9): 149-153 (November 2010). DOI: 10.3792/pjaa.86.149


In this paper we shall make complete structural elucidation of the explicit formula for the (discrete) mean square of Dirichlet $L$-function at integral arguments, save for the case $s=1$, this being completely settled in [1] recently. We shall treat the cases of negative and positive integers arguments separately, the former case being a preliminary and inclusive in the second. It will turn out that in respective cases the characteristic difference properties of Bernoulli polynomials and of the Hurwitz zeta-function are essential and telescoping the resulting difference equations, we obtain the results, revealing the underlying simple structure (known before 1905 at least).


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Guodong Liu. Nianliang Wang. Xiaohan Wang. "The discrete mean square of Dirichlet $L$-function at integral arguments." Proc. Japan Acad. Ser. A Math. Sci. 86 (9) 149 - 153, November 2010.


Published: November 2010
First available in Project Euclid: 8 November 2010

zbMATH: 1220.11102
MathSciNet: MR2780009
Digital Object Identifier: 10.3792/pjaa.86.149

Primary: 11M06
Secondary: 11M41 , 11S40

Keywords: Bernoulli polynomials , characteristic difference equations , Dirichlet $L$-function , Discrete mean square , Hurwitz zeta-function

Rights: Copyright © 2010 The Japan Academy

Vol.86 • No. 9 • November 2010
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