English

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

General Mathematics 2018-04-27 v2

Abstract

Let PP be the set of all prime numbers, q1,q2,,qmP{q_1},{q_2}, \cdots ,{q_m} \in P, PkP_k be the k-th (k=1,2,m)(k = 1,2, \cdots m) element of PP in ascending order of size, α1,α2,,αm{\alpha _1},{\alpha _2}, \cdots ,{\alpha _m} be positive integers, and β1,β2,,βm{\beta _1},{\beta _2}, \cdots ,{\beta _m} is a permutation of α1,α2,,αm{\alpha _1},{\alpha _2}, \cdots ,{\alpha _m} with β1β2βm{\beta _1} \ge {\beta _2} \ge \cdots \ge {\beta _m}, The following results are given in this paper: (i) The following inequality is true: eγloglogk=1mqkαkk=1mqk1qkαkqk1eγloglogk=1mpkβkk=1mpk1pkβkpk1{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=k=1mpkβk=(k=1mpk)1+εm(n)n = \prod\limits_{k = 1}^m {p_k^{{\beta _k}}}= {\left( {\prod\limits_{k = 1}^m {{p_k}} } \right)^{1 + {\varepsilon _m}(n)}}, limmεm(n)>0\mathop {\lim }\limits_{m \to \infty } {\varepsilon _m}(n) > 0 or limmεm(n)=+\mathop {\lim }\limits_{m \to \infty } {\varepsilon _m}(n) = + \infty, then limm(eγnloglognσ(n))>0\mathop {\lim }\limits_{m \to \infty } ({e^\gamma }n\log \log n - \sigma (n)) > 0 . Where {βk}\{ {\beta _k}\} is a sequence, βkN{\beta _k} \in N, β1β2βm{\beta _1} \ge {\beta _2} \ge \cdots \ge {\beta _m}, σ(n)=dnd\sigma (n) = \sum\limits_{\left. d \right|n} d, and γ\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 limmεm(n)=0\mathop {\lim }\limits_{m \to \infty } {\varepsilon _m}(n) = 0.

Keywords

Cite

@article{arxiv.1609.07555,
  title  = {The probability of Riemann's hypothesis being true is equal to 1},
  author = {Yuyang Zhu},
  journal= {arXiv preprint arXiv:1609.07555},
  year   = {2018}
}

Comments

25 pages