Let a ∈ (0, 1) and let Fs(a) be the periodized zeta function that is defined as Fs(a) = ∑ n−s exp(2πina) for 1, and extended to the complex plane via analytic continuation. Let sn = σn + itn, tn > 0, denote the sequence of nontrivial zeros of the Riemann zeta function in the upper halfplane ordered according to nondecreasing ordinates. We demonstrate that, assuming the Riemann Hypothesis, ...

The Riemann Zeta Function, ζ(s), is an important complex function whose behavior has implications for the distribution of the prime numbers among the natural numbers. Most notably, the still unsolved Riemann Hypothesis, which states that all non-trivial zeros of the zeta function have real part one-half, would imply the most regular distribution of primes possible in the context of current theo...

In this note concerning integrals involving the logarithm of the Riemann Zeta function, we extend some treatments given in previous pioneering works on the subject and introduce a more general set of Lorentz measures. We rst obtain two new equivalent formulations of the Riemann Hypothesis (RH). Then with a special choice of the measure we formulate the RH as a hidden symmetry , a global symmetr...

After the proof by R. Apéry of the irrationality of ζ(3) in 1976, a number of articles have been devoted to the study of Diophantine properties of values of the Riemann zeta function at positive integers. A survey has been written by S. Fischler for the Bourbaki Seminar in November 2002 [6]. Here, we review more recent results, including contributions by P. Bundschuh, S. Bruiltet, C. Elsner, S....

A simple proof of the classical subconvexity bound ζ( 1 2 + it) ≪ε t1/6+ε for the Riemann zeta-function is given, and estimation by more refined techniques is discussed. The connections between the Dirichlet divisor problem and the mean square of |ζ( 1 2 + it)| are analysed. 1. Convexity for the Riemann zeta-function Let as usual (1.1) ζ(s) = ∞

This paper uses cybernetic approach to study behavior of the Riemann zeta function. It is based on the elementary cybernetic concepts like feedback, transfer functions, time delays, PI (Proportional–Integral) controllers or FOPDT (First Order Plus Dead Time) models, respectively. An unusual dynamic interpretation of the Riemann zeta function is obtained.

We establish relationships between mean values of products of logarithmic derivatives of the Riemann zeta-function near the critical line, correlations of the zeros of the Riemann zeta-function and the distribution of integers representable as a product of a fixed number of prime powers.

In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only if n = P 1?(1?1==) n has n > 0 for n = 1; 2; 3; : : : where runs over the complex zeros of the Riemann zeta function. We show that Li's criterion follows as a consequence of a general set of inequalities for an arbitrary multiset of complex numbers and therefore is not speciic to zeta functions. We also give an a...

Preface These notes treat the problem of counting the number of rational points on a curve defined over a finite field. The notes are an extended version of an earlier set of notes Aritmetisk Algebraisk Geometri – Kurver by Johan P. Hansen [Han] on the same subject. In Chapter 1 we summarize the basic notions of algebraic geometry, especially rational points and the Riemann-Roch theorem. For th...