A smooth version of Landau´s explicit formula

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 μ is a prime number, itis sufficient to know the location of a number of nontrivial zeros of the Riemann zetafunction of order μ log 3/2 μ. Next we use Heisenberg’s inequality to support the conjecturethat this number of zeros cannot be essentially diminished.