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



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.


L-function Discrete mean value Bernoulli polynomial Clausen function 

Mathematics Subject Classification (2000)

11F66 11M26 11M41 


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Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2009

Authors and Affiliations

  1. 1.Graduate School of Advanced TechnologyKinki UniversityFukuokaJapan
  2. 2.Department of MathematicsJilin UniversityJilinPeople’s Republic of China
  3. 3.School of Mathematics and System SciencesShandong UniversityShandongPeople’s Republic of China
  4. 4.Department of MathematicsNorthwest UniversityShanxiPeople’s Republic of China

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