## 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

**Unique-IP document downloads:** 198 times

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*