On the Balazard-Saias criterion for the Riemann Hypothesis

On Balazard, Saias, and Yor's equivalence to the Riemann Hypothesis

Balazard, Saias, and Yor proved that the Riemann Hypothesis is equivalent to a certain weighted integral of the logarithm of the Riemann zeta-function along the critical line equaling zero. Assuming the Riemann Hypothesis, we investigate the rate at which a truncated version of this integral tends to zero, answering a question of Borwein, Bradley, and Crandall and disproving a conjecture of the same authors. A simple modification of our techniques gives a new proof of a classical Omega theorem for the function S(t) in the theory of the Riemann zeta-function.

Balazard--Saias--Yor准则的有限半径Jensen与相对熵形式

Finite-radius Jensen and relative-entropy formulations of the Balazard--Saias--Yor criterion

本文给出黎曼假设的两种Balazard--Saias--Yor型准则的显式重述，通过有限半径处理平凡零点并结合信息论方法，未给出黎曼假设的新证明。

A lower bound in an approximation problem involving the zeros of the Riemann zeta function

We slightly improve the lower bound of Baez-Duarte, Balazard, Landreau and Saias in the Nyman-Beurling formulation of the Riemann Hypothesis as an approximation problem. We construct Hilbert space vectors which could prove useful in the context of the so-called `Hilbert-Polya idea'.

A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis II

This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type are established; contrary to the preceding paper the focus now is on integrals involving the argument of the Riemann zeta-function (imaginary part of logarithm) rather than the logarithm of its module (real part of logarithm). Preliminary results of the numerical resear…