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2005-09

On the formulas for pi(x) and psi(x) of Riemann and von-Mangoldt

by Kunik, M..


Series: 2005-09, Preprints

MSC:
42A38 Fourier and ~Fourier-Stieltjes transforms and other transforms of Fourier type

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 derivation of Riemann's prime number formula for pi(x).
We also derive explicit formulas with the nontrivial zeros of the
zeta-function for regularizations of von Mangoldt's function psi(x).
The regularizations are based on cardinal B-splines and Gaussian
integration kernels, which are related by the Central Limit Theorem.
These results will then be generalized to a windowed Mellin or
Fourier transform with a Gaussian window function.

Keywords:
Fourier Analysis, Riemann's zeta function, Prime Numbers