Evaluation of Certain Arithmetical Functions of Number Theory & Generalization of Riemann-Weil Formula

Abstract

In this paper, we present a method to get the prime counting function π(x) and other arithmetical functions than can be generated by a Dirichlet series. First we use the general variational method to derive the solution for a Fredholm Integral equation of first kind with symmetric Kernel K(x, y)=K(y, x). We then find another integral equations with Kernels K(s, t)=K(t, s) for the Prime counting function and other  arithmetical functions generated by Dirichlet series. Additionally, we could find a solution for π(x)...for a given functional J, so that the problem of finding a formula for the density of primes on the interval [2, x] or the calculation of the coefficients for a given arithmetical function a(n) can be viewed as some “optimization” problems that can be attacked by either iterative or numerical methods. As an example we introduce Rayleigh-Ritz and Newton methods with a brief description. We also have introduced some conjectures about the asymptotic behavior of the series ...=S_n(x) for n>0 and a new expression for the Prime counting function in terms of the Non-trivial zeros of  Riemann Zeta and its connection to Riemman Hypothesis and operator theory.