Riemann hypothesis and some new integrals connected with the integral negativity of the remainder in the formula for the prime-counting function $\pi(x)$.

link: http://arxiv.org/abs/1105.4945
Abstract

In this paper a new integral for the remainder of $\pi(x)$ is obtained. It is
proved that there is an infinite set of the formulae containing miscellaneous
parts of this integral.