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