DOI QR코드

DOI QR Code

ON THE MEAN VALUES OF L(1, χ)

  • Wu, Zhaoxia (Department of Mathematics Northwest University) ;
  • Zhang, Wenpeng (Department of Mathematics Northwest University)
  • Received : 2011.06.10
  • Published : 2012.11.30

Abstract

Let $p$ > 2 be a prime, and let $k{\geq}1$ be an integer. Let ${\chi}$ be a Dirichlet character modulo $p$, and let $L(s,{\chi})$ be the Dirichlet L-function corresponding to ${\chi}$. In this paper we consider the mean values of $$\sum_{{\chi}\;mod\;p\\{\chi}(-1)=-1}{\chi}(2^k)|L(1,\chi)|^2$$.

References

  1. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
  2. T. Funakura, On Kronecker's limit formula or Dirichlet series with periodic coefficients, Acta Arith. 55 (1990), no. 1, 59-73. https://doi.org/10.4064/aa-55-1-59-73
  3. M. Katsurada and K. Matsumoto, The mean value of Dirichlet L-functions at integer points and class numbers of cyclotomic fields, Nagoya Math. J. 134 (1994), 151-172. https://doi.org/10.1017/S0027763000004906
  4. H. Liu and W. Zhang, On the mean value of L(m,${\chi}$)L(n,${\bar{\chi}}$) at positive integers m, n ${\geq}$ 1, Acta Arith. 122 (2006), no. 1, 51-56. https://doi.org/10.4064/aa122-1-5
  5. S. Louboutin, Quelques formules exactes pour des moyennes de fonctions L de Dirichlet, Canad. Math. Bull. 36 (1993), no. 2, 190-196. https://doi.org/10.4153/CMB-1993-028-8
  6. S. Louboutin, On the mean value of ${\left|L(1,{\chi})\right|}^2$ for odd primitive Dirichlet characters, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), no. 7, 143-145. https://doi.org/10.3792/pjaa.75.143
  7. S. Louboutin, The mean value of ${\left|L(\kappa,{\chi})\right|}^2$ at positive rational integers ${\geq}$ ${\geq}$ 1, Colloq. Math. 90 (2001), no. 1, 69-76. https://doi.org/10.4064/cm90-1-6
  8. H. Walum, An exact formula for an average of L-series, Illinois J. Math. 26 (1982), no. 1, 1-3.

Cited by

  1. On the mean values of Dirichlet L-functions vol.147, 2015, https://doi.org/10.1016/j.jnt.2014.07.005
  2. TWISTED QUADRATIC MOMENTS FOR DIRICHLET L-FUNCTIONS vol.52, pp.6, 2015, https://doi.org/10.4134/BKMS.2015.52.6.2095