Complex Aspects of the Riemann Hypothesis: a Computational Approach

The Riemann Zeta-function is related to the fundamental law of distribution of the primes and it is known that a very important problem in mathematics and related fields is connected with a proof of the Riemann Hypothesis (RH). The RH has connections with many areas: arithmetics, quantum theory, phase transition, dynamical systems, chaos and cryptografy. In this note we first recall some properties of the Zeta-function and briefly notice on some results reported in recent pioneering works. We then present our numerical treatment concerning the Riemann-Mangoldt function calculated on the primes up to 2.15 billions and we give a new estimation of the RiemannMangoldt constant C. Finally we develope a formal perturbation expansion and analyse it in a first order approximation. The values for the low zeroes of the Zeta-function are also given.