Values of Dirichlet L-functions, Gauss sums and trigonometric sums

Abstract

Motivated by the classical work of Ramanujan and recent work of Berndt and Zaharescu, we establish certain infinite families of identities relating values of Dirichlet L-functions at positive integers to Gauss sums and trigonometric sums twisted with characters.

Character analogue of the Boole summation formula with applications

“…, for even χ, which are analogues of Eqs. (5.9)-(5.12) of Alkan [1]. Such sums and many ones can be found in [3,7,20].…”

“…where f (l) (x) is absolutely integrable over [α, β] , and E p (x) is the Euler function defined by (1).…”

On the mean square average of special values of L-functions

“…Our main result further shows that, in an almost all sense, the upper bounds for |L(1, χ )| when χ is an odd character modulo k are better than both Littlewood's conditional upper bound for real primitive χ and the upper bounds of Bateman, Chowla and Erdös given above. The analogous problem of determining the mean square average of partial sums of primitive characters was studied earlier by Bateman and Chowla [3].…”

Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

“…The aim of this paper is to obtain a transformation formula for a very large class of Eisenstein series defined by G(z, s; A α , B β ; r 1 , r 2 ) = ∞ m,n=−∞ ′ f (αm)f * (βn) ((m + r 1 )z + n + r 2 ) s , Re(s) > 2, Im(z) > 0 (2) where {f (n)} and {f * (n)} , −∞ < n < ∞ are sequences of complex numbers with period k > 0, and A α = {f (αn)} and B β = {f * (βn)} . In (2), the dash ′ means that the possible pair m = −r 1 , n = −r 2 is excluded from the summation. Generalizations of Dedekind sums involving the periodic Bernoulli function appear in the transformation formulae.…”

“…The definition of the Eisenstein series in (2) and the methods presented in the sequel are motivated by [6] and [10].…”