The Riemann Hypothesis, which specifies the location of zeros of the Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems in mathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of cases. Recently a new algorithm, invented by the speaker and A. Scho . . nhage, has been implement...

We expose techniques that permit to approximate the Riemann Zeta function in different context. The motivation does not restrict only to the numerical verification of the Riemann hypothesis (see Numerical computations about the zeros of the zeta function) : in [5] for example, Lagarias and Odlyzko obtained the best asymptotic known method to compute π(x), the number of primes less than x, thank...

The “hybrid” moments Z 2T T |ζ( 1 2 + it)| „ Z t+G t−G |ζ( 1 2 + ix)| dx m dt of the Riemann zeta-function ζ(s) on the critical line Re s = 1 2 are studied. The expected upper bound for the above expression is Oε(T G). This is shown to be true for certain specific values of k, l,m ∈ N, and the explicitly determined range of G = G(T ; k, l,m). The application to a mean square bound for the Melli...

Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an elementary combinatorial argument, we prove that assuming the Riemann Hypothesis at least 67.275% of the zeros are simple.

According to my first teacher Gustave Choquet one does, by openly facing a well known unsolved problem, run the risk of being remembered more by one’s failure than anything else. After reaching a certain age, I realized that waiting “safely” until one reaches the end-point of one’s life is an equally selfdefeating alternative. In this paper I shall first look back at my early work on the classi...

In recent years there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications. PACS numbers: 02.10.De, 02.10.Yn 1. The history in brief Number theory and random matrix theory met, by chance, over a cup of tea in...