On the formulas for Pi(​x) and Psi(x) of Riemann and von Mangoldt

Abstract

Using the Mellin transform and the complex exponential integral we derive various representation formulas for the factors of the entire functions in Hadamards product theorem. The application of these results on Riemann’s zeta function leads to a new derivation of Rie- mann’s prime number formula for Pi(x). We will thereby present a correct version of this formula, which is given in a wrong way in the literature. Using the nontrivial zeros of the Zeta function we also obtain explicit formulas for regularizations of von Mangoldt’s function Psi(x). These regularizations are based on cardinal B-splines and Gaussian integration kernels, which are related by the Central Limit Theorem. Our results will then be generalized to a windowed Mellin or Fourier transform with a Gaussian window function.