Geodesic
Explicit formula for sums related to the generalized Bernoulli numbers
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 1, pp. 135-141.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\chi$ be a Dirichlet character modulo a prime number $p\geqslant 3$ and let $B_m(\chi)$ $(m=1,2,\ldots)$ be the generalized Bernoulli numbers associated with $\chi$. Explicit formulas for the sums: $$\sum_{\substack{\chi\mod p\\\chi(-1)=+1, \chi\neq\chi_0}}B_{m}(\chi)B_{n}(\overline{\chi})\text{ and }\sum_{\substack{\chi\mod p\\ \chi(-1)=-1}}B_{m}(\chi)B_{n}(\overline{\chi})$$ are given in this paper.
Keywords: character sum, Dirichlet $L$-function, Bernoulli number, generalized Bernoulli number. 
@article{JSFU_2023_16_1_a12,
     author = {Brahim Mittou},
     title = {Explicit formula for sums related to the generalized {Bernoulli} numbers},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {135--141},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2023_16_1_a12/}
}
TY  - JOUR
AU  - Brahim Mittou
TI  - Explicit formula for sums related to the generalized Bernoulli numbers
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2023
SP  - 135
EP  - 141
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2023_16_1_a12/
LA  - en
ID  - JSFU_2023_16_1_a12
ER  - 
%0 Journal Article
%A Brahim Mittou
%T Explicit formula for sums related to the generalized Bernoulli numbers
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2023
%P 135-141
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2023_16_1_a12/
%G en
%F JSFU_2023_16_1_a12
Brahim Mittou. Explicit formula for sums related to the generalized Bernoulli numbers. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 1, pp. 135-141. http://geodesic.mathdoc.fr/item/JSFU_2023_16_1_a12/

[1] T.M Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976

[2] T.Arakawa, T.Ibukiyama, M.Kaneko, Bernoulli Numbers and Zeta Functions, Springer Japan, 2014

[3] K.-W.Chen, M.Eie, “A note on generalized Bernoulli numbers”, Pacific J. Math., 1 (2001), 41–59

[4] H.Liu, “On the mean values of Dirichlet L-function”, J. Number Theory, 147 (2015), 172–183

[5] H.Liu, W.Zhang, “On the mean value of $L(m,\chi)L(n,\chi)$ at positive integers $m,n\geq1$”, Acta Arith., 122 (2006), 51–56

[6] S.Louboutin, “Twisted quadratic moments for Dirichlet $L$-functions at $s=2$”, Publ. Math. Debrecen, 95 (2019), 393–400

[7] B.Mittou, A.Derbal, “Complex numbers similar to the generalized Bernoulli numbers and their applications”, Math. Montisnigri, L (2021), 15–26 | DOI