## On the Evaluation of Certain Arithmetical Functions of Number Theory and Their Sums

**Authors:** Jose Javier Garcia Moreta

ABSTRACT: In this paper we present a method to get the prime counting function (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), after that we 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, then we could find a solution for (x) and ( ) ( )
n x
a n A x
, solving J[ ] 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)
Also we have introduced some conjectures about the asymptotic behavior of the series
( ) n
n
p x
x p
=Sn(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.
Keywords: =PNT (prime

**Comments:** 11 Pages.

**Download:** **PDF**

### Submission history

[v1] 29 Jan 2010

[v2] 2014-04-19 05:49:02

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