# Number Theory

## 1001 Submissions

[3] **viXra:1001.0047 [pdf]**
*submitted on 29 Jan 2010*

### An Approach to π(x) and Other Arithmetical Functions by Variational Principles

**Authors:** Jose Javier Garcia Moreta

**Comments:** 11 Pages.

In this paper we present a method to get the prime counting function p(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 ... (see paper for full abstract)

**Category:** Number Theory

[2] **viXra:1001.0039 [pdf]**
*submitted on 26 Jan 2010*

### A Comment on Mathematical Methods to Deal with Divergent Series and Integrals

**Authors:** Jose Javier Garcia Moreta

**Comments:** 14 Pages.

In this paper we study the methods of Borel and Nachbin resummation applied
to the solution of integral equation with Kernels K(yx) , the resummation of divergent series
and the possible application to Hadamard finite-part integral and distribution theory.

**Category:** Number Theory

[1] **viXra:1001.0038 [pdf]**
*replaced on 7 Mar 2010*

### A Note on the Mellin Convolution of Functions and Its Relation to Riesz Criterion and Riemann Hypothesis

**Authors:** Jose Javier Garcia Moreta

**Comments:** 8 Pages.

In this paper we study how the Mellin convolution of functions f and g
( f * g ) and how is related to the Riesz criterion for the Riemann Hypothesis, the idea
is to stablish a Fredholm integral equation of First kind for the Riesz function and the
Hardy function.

**Category:** Number Theory