Geodesic
On the mean value of the generalized Dirichlet $L$-functions
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 597-620 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $s=\frac 12+{\rm i} t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.
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 $s=\frac 12+{\rm i} t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.
Classification : 11M20
Keywords: generalized Dirichlet $L$-functions; mean value properties; functional equation; asymptotic formula 
@article{CMJ_2010_60_3_a1,
     author = {Ma, Rong and Yi, Yuan and Zhang, Yulong},
     title = {On the mean value of the generalized {Dirichlet} $L$-functions},
     journal = {Czechoslovak Mathematical Journal},
     pages = {597--620},
     year = {2010},
     volume = {60},
     number = {3},
     mrnumber = {2672404},
     zbl = {1224.11077},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a1/}
}
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
SP  - 597
EP  - 620
VL  - 60
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a1/
LA  - en
ID  - CMJ_2010_60_3_a1
ER  - 
%0 Journal Article
%A Ma, Rong
%A Yi, Yuan
%A Zhang, Yulong
%T On the mean value of the generalized Dirichlet $L$-functions
%J Czechoslovak Mathematical Journal
%D 2010
%P 597-620
%V 60
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a1/
%G en
%F CMJ_2010_60_3_a1
Ma, Rong; Yi, Yuan; Zhang, Yulong. On the mean value of the generalized Dirichlet $L$-functions. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 597-620. http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a1/