In this paper we extend to arbitrary number fields a construction of Bost-Connes of a -dynamical system with spontaneous symmetry breaking and partition function the Riemann zeta function.
We introduce a family of deformations of the Riemann xi-function endowed with two continuous parameters. We show that it has rich analytic structure and that its conjectural (mild) zero-free region for some fixed parameter is a sufficient condition for the Riemann hypothesis to hold for the Riemann zeta function.
We generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give new integral representations of several zeta functions, an extension of the parity result to the whole domain of convergence, concrete expressions of Tornheim's double zeta function at non-positive integers and some results on the behavior of a certain Witten's zeta function...
We give higher-power generalizations of the classical Lerch formula for the gamma function.
We generalize Rademacher's reciprocity formula for the Dedekind sum to a family of cotangent sums. One of the sums in this family is strictly related to the Vasyunin sum, a function defined on the rationals that is relevant to the Nyman-Beurling-Báez-Duarte approach to the Riemann hypothesis.
The main purpose of this paper is to use the M. Toyoizumi's important work, the properties of the Dedekind sums and the estimates for character sums to study a hybrid mean value of the Dedekind sums, and give a sharper asymptotic formula for it.
The main purpose of this paper is using the mean value formula of Dirichlet L-functions and the analytic methods to study a hybrid mean value problem related to certain Hardy sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.