05-09
On the formulas for pi(x) and psi(x) of Riemann and von-Mangoldt
by Kunik, M.
Preprint series: 05-09, 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
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