Number Theory


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.

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Submission history

[v1] 29 Jan 2010
[v2] 2014-04-19 05:49:02

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