The Complete Proof of the Riemann Hypothesis

EasyChair Preprint 6710, version 2

Abstract

Robin criterion states that the Riemann Hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We prove that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $2$ and $953$. Using this result, we show there is a contradiction just assuming the possible smallest counterexample $n > 5040$ of the Robin inequality. In this way, we prove that the Robin inequality is true for all $n > 5040$ and thus, the Riemann Hypothesis is true.

Keyphrases: Riemann hypothesis, Robin inequality, prime numbers, sum-of-divisors function

@booklet{EasyChair:6710,
  author    = {Frank Vega},
  title     = {The Complete Proof of the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 6710},
  year      = {EasyChair, 2021}}