An explicit formula for the zeros of the Riemann zeta function

Abstract

We present an explicit formula for a weighted sum over the zeros of the Riemann zeta function. This weighted sum is evaluated in terms of a sum over the prime numbers, weighted with help of the Hermite polynomials. From the explicit formula presented in this note, it follows that prime numbers determine the distribution of the zeros of the Riemann zeta function. In fact, it is possible to compute the zeros of the zeta function without actually using this function.