On the discrete mean value of the product of two Dirichlet L-functions

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Abstract

In Acta. Arith. 122(1), 51–56, 2006, Liu and the third author elaborated on the result of Louboutin to determine the coefficients of the exact formula for the discrete mean value \(\sum_{{\scriptstyle\chi\bmod q}\atop {\scriptstyle\chi(-1)=(-1)^{m}}}L(m,\chi)L(n,\bar{\chi})\) of the product of two Dirichlet L-functions, where m,n and χ are of the same parity. The method uses the Fourier series for the periodic Bernoulli polynomials and is rather computational. In this paper, we shall reveal the hidden algebraic structures and the intrinsic properties of the Bernoulli polynomials and of the Clausen functions to treat the more difficult case of m and χ being of opposite parity. The additive group structure of ℤ/qℤ is realized as the discrete Fourier transform while the multiplicative group structure of (ℤ/qℤ)× is realized as the subgroup of all even characters in the character group \(\widehat{(\mathbb{Z}/q\mathbb{Z})^{\times}}\) . The intrinsic property is the distribution property, which corresponds to the equally divided Riemann sum. This suggests the analogy between our result and continuous mean values in the form of definite integrals.

Communicated by V. Cortés.
Dedicated to Professor Yoshiyuki Kitaoka on his sixtieth birthday with great respect and friendship!
Jing Ma supported in part by China NNSF (10726051) and Grant in-aid for Scientific Research from Department of Mathematics, Jilin University.