Asymptotic harmonic behavior in the prime number distribution

TL;DR

This paper explores a function related to prime numbers and shows that its boundedness is connected to the Riemann hypothesis or the existence of infinitely many zeros of the zeta function with real part greater than one-half, supported by numerical evidence up to one trillion.

Contribution

It introduces a new function involving primes whose boundedness relates to the Riemann hypothesis and provides numerical analysis of its harmonic behavior based on the first 21 zeros.

Findings

Numerical evidence of harmonic behavior up to one trillion.

Boundedness of the function implies either the Riemann hypothesis or infinitely many zeros with real part > 1/2.

Connection between prime sums and zeros of the zeta function.

Abstract

We consider $\Phi(x)=x^{-\frac{1}{4}}\left[1-2\sqrt{x}\Sigma e^{-p^2\pi x}\ln p\right]$ on $x>0$, where the sum is over all primes $p$. If $\Phi$ is bounded on $x>0$, then the Riemann hypothesis is true or there are infinitely many zeros Re~$z_k>\frac{1}{2}$. The first 21 zeros give rise to asymptotic harmonic behavior in $\Phi(x)$ defined by the prime numbers up to one trillion.