The probability of Riemann's hypothesis being true is equal to 1

TL;DR

This paper presents new inequalities related to prime numbers and their products, and concludes that the probability of Riemann's hypothesis being true is 1 based on these mathematical results.

Contribution

The paper introduces novel inequalities involving prime products and their sums, and claims to prove that Riemann's hypothesis is almost surely true.

Findings

Inequalities involving prime products are established.

Conditions under which certain limits are positive are identified.

It concludes the probability of Riemann's hypothesis being true is 1.

Abstract

Let $P$ be the set of all prime numbers, ${q_1},{q_2}, \cdots ,{q_m} \in P$, $P_k$ be the k-th $(k = 1,2, \cdots m)$ element of $P$ in ascending order of size, ${\alpha _1},{\alpha _2}, \cdots ,{\alpha _m}$ be positive integers, and ${\beta _1},{\beta _2}, \cdots ,{\beta _m}$ is a permutation of ${\alpha _1},{\alpha _2}, \cdots ,{\alpha _m}$ with ${\beta _1} \ge {\beta _2} \ge \cdots \ge {\beta _m}$, The following results are given in this paper: (i) The following inequality is true: ${e^\gamma }\log \log \prod\limits_{k = 1}^m {q_k^{{\alpha _k}}} - \prod\limits_{k = 1}^m {\frac{{{q_k} - {\textstyle{1 \over {q_k^{{\alpha _k}}}}}}}{{{q_k} - 1}}} \ge {e^\gamma }\log \log \prod\limits_{k = 1}^m {p_k^{{\beta _k}}} - \prod\limits_{k = 1}^m {\frac{{{p_k} - {\textstyle{1 \over {p_k^{{\beta _k}}}}}}}{{{p_k} - 1}}}$. (ii) If $n = \prod\limits_{k = 1}^m {p_k^{{\beta _k}}}= {\left( {\prod\limits_{k = 1}^m {{p_k}} } \right)^{1 + {\varepsilon _m}(n)}}$, $\mathop {\lim }\limits_{m \to \infty } {\varepsilon _m}(n) > 0$ or $\mathop {\lim }\limits_{m \to \infty } {\varepsilon _m}(n) = + \infty$, then $\mathop {\lim }\limits_{m \to \infty } ({e^\gamma }n\log \log n - \sigma (n)) > 0$ . Where $\{ {\beta _k}\}$ is a sequence, ${\beta _k} \in N$, ${\beta _1} \ge {\beta _2} \ge \cdots \ge {\beta _m}$, $\sigma (n) = \sum\limits_{\left. d \right|n} d$, and $\gamma$ is the Euler constant. (iii) The probability of Riemann's hypothesis being true is equal to 1. In addition, two results are given when $\mathop {\lim }\limits_{m \to \infty } {\varepsilon _m}(n) = 0$.