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The Riemann Hypothesis

EasyChair Preprint 3708, version 50

5 pagesDate: June 15, 2021

Abstract

Let's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}-\log \log x-B)$, where $B \approx 0.2614972128$ is the Meissel-Mertens constant. The Robin theorem states that $\delta(x)$ changes sign infinitely often. Let's also define $S(x) = \theta(x) - x$, where $\theta(x)$ is the Chebyshev function. It is known that $S(x)$ changes sign infinitely often. Using the Nicolas theorem, we prove that when the inequalities $\delta(x) \leq 0$ and $S(x) \geq 0$ are satisfied for some number $x \geq 127$, then the Riemann Hypothesis should be false. However, the Mertens second theorem states that $\lim_{{x\to \infty }} \delta(x) = 0$. Moreover, we know that $\lim_{{x\to \infty }} S(x) = 0$. 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

BibTeX entry
BibTeX does not have the right entry for preprints. This is a hack for producing the correct reference:
@booklet{EasyChair:3708,
  author    = {Frank Vega},
  title     = {The Riemann Hypothesis},
  howpublished = {EasyChair Preprint 3708},
  year      = {EasyChair, 2021}}
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