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On discrete mean values of Dirichlet L -functions

Ertan Elma (2021)

Czechoslovak Mathematical Journal

Let χ be a nonprincipal Dirichlet character modulo a prime number p 3 and let 𝔞 χ : = 1 2 ( 1 - χ ( - 1 ) ) . Define the mean value p ( - s , χ ) : = 2 p - 1 ψ ( mod p ) ψ ( - 1 ) = - 1 L ( 1 , ψ ) L ( - s , χ ψ ¯ ) ( σ : = s > 0 ) . We give an identity for p ( - s , χ ) which, in particular, shows that p ( - s , χ ) = L ( 1 - s , χ ) + 𝔞 χ 2 p s L ( 1 , χ ) ζ ( - s ) + o ( 1 ) ( p ) for fixed 0 < σ < 1 2 and | t : = s | = o ( p ( 1 - 2 σ ) / ( 3 + 2 σ ) ) .

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