Evaluation of Certain Arithmetical Functions of Number Theory & Their Sums

Abstract

In this paper, we present a method to get the prime counting function p(x) and other arithmetical functions beyond what 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.  We could find a solution for p(x) and åa(n)=A(x) with n £ x, solving dJ[f]=0 for a given functional J, so 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. Further, we will introduce some conjectures about the asymptotic behavior of the series X_n(x) = åpⁿ = S_n(x) for p £ x and 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.