The probability of Riemann's hypothesis being true is equal to 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: $e^{\gamma }\log \log \prod _{k=1}^m{q}_k^{{\alpha }_k}-\prod _{k=1}^m\frac{q_k-\frac{1}{q_k^{{\alpha }_k}}}{q_k-1}\ge e^{\gamma }\log \log \prod _{k=1}^m{p}_k^{{\beta }_k}-\prod _{k=1}^m\frac{p_k-\frac{1}{p_k^{{\beta }_k}}}{p_k-1}$. (ii) If $n=\prod _{k=1}^m{p}_k^{{\beta }_k}={\left(\prod _{k=1}^m{p}_k\right)}^{1+{\epsilon }_m(n)}$, $\underset{m\to \infty }{\lim }{\epsilon }_m(n)>0$ or $\underset{m\to \infty }{\lim }{\epsilon }_m(n)=+\infty$, then $\underset{m\to \infty }{\lim }(e^{\gamma }n\log \log n-\sigma (n))>0$ . Where $\{{\beta }_k\}$ is a sequence, ${\beta }_k\in N$, ${\beta }_1\ge {\beta }_2\ge \cdots \ge {\beta }_m$, $\sigma (n)=\sum _{d|n}d$, 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 $\underset{m\to \infty }{\lim }{\epsilon }_m(n)=0$.

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