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

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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. 
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     title = {Explicit formula for sums related to the generalized {Bernoulli} numbers},
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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/