The Riemann Hypothesis

EasyChair Preprint 3708, version 47

Abstract

Let's define $\delta(n) = (\sum_{{q\leq n}}{\frac{1}{q}}-\log \log n-B)$, where $B \approx 0.2614972128$ is the Meissel-Mertens constant. The Robin theorem states that $\delta(n)$ changes sign infinitely often. We prove if the inequality $\delta(p) \leq 0$ holds for a prime $p$ big enough, then the Riemann Hypothesis should be false. However, we could restate the Mertens second theorem as $\lim_{{n\to \infty }} \delta(p_{n}) = 0$ where $p_{n}$ is the $n^{th}$ prime number. In this way, this work could mean a new step forward in the direction for finally solving the Riemann Hypothesis.

Keyphrases: Divisor, inequality, number theory

@booklet{EasyChair:3708,
  author    = {Frank Vega},
  title     = {The Riemann Hypothesis},
  howpublished = {EasyChair Preprint 3708},
  year      = {EasyChair, 2021}}