Type: Article

A smooth version of Landau's explicit formula

Balanzario, Eugenio P.; Cárdenas-Romero, Daniel E.; Chacón-Serna, Richar;

Abstract:

In this paper, we present a smooth version of Landau’s explicit formula for the von Mangoldt arithmetical function. By assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number ? is a prime number, it is sufficient to know the location of a number of nontrivial zeros of the Riemann zeta function of order ?log32?. Next we use Heisenberg’s inequality to support the conjecture that this number of zeros cannot be essentially diminished.