| Title:
|
On discrete mean values of Dirichlet $L$-functions (English) |
| Author:
|
Elma, Ertan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
1035-1048 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\chi $ be a nonprincipal Dirichlet character modulo a prime number $p\geqslant 3$ and let $\mathfrak a_\chi := \tfrac 12 (1-\chi (-1))$. Define the mean value $$ \mathcal {M}_{p}(-s,\chi ) :=\frac {2}{p-1} \sum \psi \pmod p \psi (-1)=-1 L(1,\psi )L(-s,\chi \bar {\psi }) \quad (\sigma :=\Re s>0). $$ We give an identity for $\mathcal {M}_{p}(-s,\chi )$ which, in particular, shows that $$ \mathcal {M}_{p}(-s,\chi )= L(1-s,\chi )+\mathfrak a_\chi 2p^s L(1,\chi )\zeta (-s) +o(1) \quad (p\rightarrow \infty ) $$ for fixed $0<\sigma <\frac {1}{2}$ and $|t:=\Im s|=o (p^{(1-2\sigma )/(3+2\sigma )})$. (English) |
| Keyword:
|
Dirichlet $L$-function |
| Keyword:
|
mean value |
| Keyword:
|
Dirichlet character |
| MSC:
|
11L40 |
| MSC:
|
11M06 |
| idZBL:
|
Zbl 07442472 |
| idMR:
|
MR4339109 |
| DOI:
|
10.21136/CMJ.2021.0189-20 |
| . |
| Date available:
|
2021-11-08T16:00:00Z |
| Last updated:
|
2024-01-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149236 |
| . |
| Reference:
|
[1] Davenport, H.: Multiplicative Number Theory.Graduate Texts in Mathematics 74. Springer, New York (2000). Zbl 1002.11001, MR 1790423, 10.1007/978-1-4757-5927-3 |
| Reference:
|
[2] Elma, E.: On a problem related to discrete mean values of Dirichlet $L$-functions.J. Number Theory 217 (2020), 36-43. Zbl 07242300, MR 4140619, 10.1016/j.jnt.2020.05.019 |
| Reference:
|
[3] Ivić, A.: The Riemann Zeta-Function: Theory and Applications.Dover Publications, Mineola (2003). Zbl 1034.11046, MR 1994094 |
| Reference:
|
[4] Kanemitsu, S., Ma, J., Zhang, W.: On the discrete mean value of the product of two Dirichlet $L$-functions.Abh. Math. Semin. Univ. Hamb. 79 (2009), 149-164. Zbl 1259.11076, MR 2545597, 10.1007/s12188-009-0016-1 |
| Reference:
|
[5] Liu, H., Zhang, W.: On the mean value of $L(m,\chi)L(n,\overline{\chi})$ at positive integers $m,n\geq 1$.Acta Arith. 122 (2006), 51-56. Zbl 1108.11062, MR 2217323, 10.4064/aa122-1-5 |
| Reference:
|
[6] Louboutin, S.: Quelques formules exactes pour des moyennes de fonctions $L$ de Dirichlet.Can. Math. Bull. 36 (1993), 190-196. Zbl 0802.11032, MR 1222534, 10.4153/CMB-1993-028-8 |
| Reference:
|
[7] Louboutin, S.: The mean value of $|L(k,\chi)|^2$ at positive rational integers $k\geq 1$.Colloq. Math. 90 (2001), 69-76. Zbl 1013.11049, MR 1874365, 10.4064/cm90-1-6 |
| Reference:
|
[8] Matsumoto, K.: Recent developments in the mean square theory of the Riemann zeta and other zeta-functions.Number Theory Trends in Mathematics. Birkhäuser, Basel (2000), 241-286. Zbl 0959.11036, MR 1764806, 10.1007/978-93-86279-02-6_14 |
| Reference:
|
[9] Montgomery, H. L., Vaughan, R. C.: Multiplicative Number Theory. I. Classical Theory.Cambridge Studies in Advanced Mathematics 97. Cambridge University Press, Cambridge (2007). Zbl 1142.11001, MR 2378655, 10.1017/CBO9780511618314 |
| Reference:
|
[10] Motohashi, Y.: A note on the mean value of the zeta and $L$-functions. I.Proc. Japan Acad., Ser. A 61 (1985), 222-224. Zbl 0573.10027, MR 0816718, 10.3792/pjaa.61.222 |
| Reference:
|
[11] Titchmarsh, E. C.: The Theory of the Riemann Zeta-Function.Oxford Science Publications. Clarendon Press, Oxford (1986). Zbl 0601.10026, MR 0882550 |
| Reference:
|
[12] Xu, Z., Zhang, W.: Some identities involving the Dirichlet $L$-function.Acta Arith. 130 (2007), 157-166. Zbl 1154.11027, MR 2357653, 10.4064/aa130-2-5 |
| . |
