Prime Number Diffeomorphisms, Diophantine Equations and the Riemann Hypothesis

TL;DR

This paper constructs smooth, invertible functions related to prime numbers to analyze prime distribution and Diophantine equations, potentially offering new insights into the Riemann hypothesis and prime number theorem error bounds.

Contribution

It introduces a novel smooth diffeomorphism pair based on prime series and spline interpolation, providing a new tool for prime analysis and Diophantine approximation.

Findings

Constructed explicit smooth prime-related functions

Demonstrated use in approximating Diophantine solutions over primes

Discussed potential implications for the Riemann hypothesis

Abstract

We explicitly construct a diffeomorphic pair (p(x),p^{-1}(x)) in terms of an appropriate quadric spline interpolating the prime series. These continuously differentiable functions are the smooth analogs of the prime series and the prime counting function, respectively, and contain the basic information about the specific behavior of the primes. We employ p^{-1}(x) to find approximate solutions of Diophantine equations over the primes and discuss how this function could eventually be used to analyze the von Koch estimate for the error in the prime number theorem which is known to be equivalent to the Riemann hypothesis.