Submitted:
06 June 2020
Posted:
08 June 2020
Read the latest preprint version here
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.
Keywords:
Riemann hypothesis
; Nicolas theorem
; prime numbers
; Chebyshev function
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
