An Approach to PI(x) and other Arithmetical function by Variational principles

Abstract

In this paper we present a method to derive Pi(x) and other Arithemtical functions that can be generated by a Dirichlet series by variational principles,we use a variational method to determine the solution for a Fredholm integral equation of second kind, after that we propose (obtain) two integral equations one for the Pi(x) and other for the arithmetical function A(x)=Sum(n,x)a(n) so they can be solved by usual optimization method. Also some conjectures on the value for the asymptotic value of the sum of f(t)=t^{n} are given in the form Li(x^{n+1}) Changes: Rayleigh-ritz Variational Methods added, we have also included a brief description of how to accelerate the convergence of the series Sum{p}f(x),Grammar changes.