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Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions

Abdelmejid Bayad, Yilmaz Simsek (2011)

Annales de l’institut Fourier

In this paper we study three new shifted sums of Apostol-Dedekind-Rademacher type. The first sums are written in terms of Jacobi modular forms, and the second sums in terms of cotangent and the third sums are expressed in terms of special values of the Barnes multiple zeta functions. These sums generalize the classical Dedekind-Rademacher sums. The main aim of this paper is to state and prove the Dedekind reciprocity laws satisfied by these new sums. As an application of our Dedekind reciprocity...

Dirichlet series induced by the Riemann zeta-function

Jun-ichi Tanaka (2008)

Studia Mathematica

The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on ω , the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form ( a p , s ) = ∏ p ( 1 - a p p - s ) - 1 for a p in ω . Among other things, using the Haar measure on ω for measuring the asymptotic behavior of ζ(s) in the critical strip, we shall prove, in a weak sense, the mean-value theorem for ζ(s), equivalent to the Lindelöf hypothesis.

Discrete limit theorems for general Dirichlet series. III

A. Laurinčikas, R. Macaitienė (2004)

Open Mathematics

Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general Dirichlet series. The explicit form of the limit measure in this theorem is given.

Distribution of values of Hecke characters of infinite order

C. S. Rajan (1998)

Acta Arithmetica

We show that the number of primes of a number field K of norm at most x, at which the local component of an idele class character of infinite order is principal, is bounded by O(x exp(-c√(log x))) as x → ∞, for some absolute constant c > 0 depending only on K.

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