New generating and counting Functions of prime numbers applied to approximate Chebyschev 2nd class function and the least action principle applied to find non-trivial roots of the Zeta function and to Riemann Hypothesis

TL;DR

This paper introduces new elementary functions related to prime numbers, providing analytic tools for prime counting, approximations of key functions, and a novel variational method to find non-trivial roots of the Riemann zeta function, supporting the Riemann Hypothesis.

Contribution

The paper presents a new set of prime number functions defined analytically, along with applications to prime counting, approximations of the Riemann zeta function, and a novel variational approach to locate its non-trivial roots.

Findings

New analytic prime generating and discriminating functions introduced.

Derived accurate approximations for prime counting and Riemann zeta functions.

Applied variational calculus to find and confirm non-trivial roots of the Riemann zeta function.

Abstract

We introduce a new set of prime numbers functions including an exact Generating Function and a Discriminating Function of Prime Numbers neither based on prime number tables nor on algorithms. Instead these functions are defined in terms of ordinary elementary functions, therefore having the advantage of being analytic and readily calculable. Also presented are four applications of our new Prime Numbers Generating Function, namely: obtaining a new analytic formula for counting prime numbers, obtaining an approximant to Euler product function, obtaining an approximant to Riemann Zeta (sigma,tau) function based on our primes discriminating function, an accurate approximant to the Chebyshev function of second class in terms of our primes generating function, and the application of this approximant in sharp estimates related to the validity of the Riemann Hypothesis. We also apply the variational calculus of classical mechanics to obtain the non-trivial roots of Riemann zeta function in the complex plane, in an original and novel approach. A variational test function based on the modulus squared of Riemann function is defined, and then Hamilton Principle is applied to analytically obtain the non-trivial roots of Riemann zeta in a completely original way. We optimize our analytical procedure by defining a more general test function that depends explicitly on the abscissa variable sigma, and present a procedure to find non-trivial roots along the critical line, as demanded by Riemann Hypothesis, thus confirming it. Our method even allows us to define a function, that behaves analogous to the Riemann zeta function, and that even admits a functional type equation.