| Title:
|
On the mean value of the generalized Dirichlet $L$-functions (English) |
| Author:
|
Ma, Rong |
| Author:
|
Yi, Yuan |
| Author:
|
Zhang, Yulong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
3 |
| Year:
|
2010 |
| Pages:
|
597-620 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
generalized Dirichlet $L$-functions |
| Keyword:
|
mean value properties |
| Keyword:
|
functional equation |
| Keyword:
|
asymptotic formula |
| MSC:
|
11M20 |
| idZBL:
|
Zbl 1224.11077 |
| idMR:
|
MR2672404 |
| . |
| Date available:
|
2010-07-20T17:03:12Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140593 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
