On the mean value of the generalized Dirichlet L -functions

Rong Ma; Yuan Yi; Yulong Zhang

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 3, page 597-620
  • ISSN: 0011-4642

Abstract

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Let q ≥ 3 be an integer, let χ denote a Dirichlet character modulo q . For any real number a ≥ 0 we define the generalized Dirichlet L -functions L ( s , χ , a ) = ∑ n = 1 ∞ χ ( n ) ( n + a ) s , where s = σ + i t with σ > 1 and t both real. They can be extended to all s by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet L -functions especially for s = 1 and s = 1 2 + i t , and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.

How to cite

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Ma, Rong, Yi, Yuan, and Zhang, Yulong. "On the mean value of the generalized Dirichlet $L$-functions." Czechoslovak Mathematical Journal 60.3 (2010): 597-620. <http://eudml.org/doc/38030>.

@article{Ma2010,
abstract = {Let $q\ge 3$ be an integer, let $\chi $ denote a Dirichlet character modulo $q.$ For any real number $a\ge 0$ we define the generalized Dirichlet $L$-functions \[ L(s,\chi ,a)=\sum \_\{n=1\}^\{\infty \}\frac\{\chi (n)\}\{(n+a)^s\}, \] where $s=\sigma +\{\rm i\} t$ with $\sigma >1$ and $t$ both real. They can be extended to all $s$ by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet $L$-functions especially for $s=1$ and $s=\frac\{1\}\{2\}+\{\rm i\} t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.},
author = {Ma, Rong, Yi, Yuan, Zhang, Yulong},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized Dirichlet $L$-functions; mean value properties; functional equation; asymptotic formula; generalized Dirichlet -function; mean value property; functional equation; asymptotic formula},
language = {eng},
number = {3},
pages = {597-620},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the mean value of the generalized Dirichlet $L$-functions},
url = {http://eudml.org/doc/38030},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Ma, Rong
AU - Yi, Yuan
AU - Zhang, Yulong
TI - On the mean value of the generalized Dirichlet $L$-functions
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 3
SP - 597
EP - 620
AB - Let $q\ge 3$ be an integer, let $\chi $ denote a Dirichlet character modulo $q.$ For any real number $a\ge 0$ we define the generalized Dirichlet $L$-functions \[ L(s,\chi ,a)=\sum _{n=1}^{\infty }\frac{\chi (n)}{(n+a)^s}, \] where $s=\sigma +{\rm i} t$ with $\sigma >1$ and $t$ both real. They can be extended to all $s$ by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet $L$-functions especially for $s=1$ and $s=\frac{1}{2}+{\rm i} t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.
LA - eng
KW - generalized Dirichlet $L$-functions; mean value properties; functional equation; asymptotic formula; generalized Dirichlet -function; mean value property; functional equation; asymptotic formula
UR - http://eudml.org/doc/38030
ER -

References

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  7. Yi, Y., Zhang, W. P., On the 2 k -th power mean of Dirichlet L -functions with the weight of Gauss sums, Advances in Mathematics 31 (2002), 517-526. (2002) MR1959549
  8. Balasubramanian, R., 10.4064/aa-38-3-273-283, Acta Arith. 38 (1980), 273-283. (1980) MR0602193DOI10.4064/aa-38-3-273-283
  9. Titchmarsh, E. C., The Theory of the Riemannn Zeta-function, Oxford (1951). (1951) MR0046485
  10. Ivic, A., The Riemann zeta-function, The Theory of the Riemann Zeta-Function with Applications, New York: Wiley (1985). (1985) Zbl0583.10021MR0792089
  11. Pan, C. D., Pan, C. B., Elements of the Analytic Number Theory, Science Press, Beijing (1991), Chinese. (1991) 

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