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

2016/09/24 by Yuyang Zhu, Zhu, Yuyang
Mathematics · Physics and Astronomy · #11M26 #Advanced Algebra and Geometry #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM)

paper · pdf · doi:10.48550/arxiv.1609.07555

openalex publication_date 2016/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P be the set of all prime numbers, q₁,q₂, ⋯ ,q_m ∈ P, P_k be the k-th (k = 1,2, ⋯ m) element of P in ascending order of size, α₁,α₂, ⋯ ,α_m be positive integers, and β₁,β₂, ⋯ ,β_m is a permutation of α₁,α₂, ⋯ ,α_m with β₁ ≥ β₂ ≥ ⋯ ≥ β_m, The following results are given in this paper: (i) The following inequality is true: e^γlog log ∏_{k = 1}^m q_k^α_k - ∏_{k = 1}^m \fracq_k - \textstyle1 \over q_k^α_kq_k - 1 ≥ e^γlog log ∏_{k = 1}^m p_k^β_k - ∏_{k = 1}^m \fracp_k - \textstyle1 \over p_k^β_kp_k - 1. (ii) If n = ∏_{k = 1}^m p_k^β_k= ( ∏_{k = 1}^m p_k )^1 + ε _m(n), \mathop lim _{m → ∞} ε _m(n) > 0 or \mathop lim _{m → ∞} ε _m(n) = + ∞, then \mathop lim _{m → ∞} (e^γnlog log n - σ(n)) > 0 . Where \ β_k\ is a sequence, β_k ∈ N, β₁ ≥ β₂ ≥ ⋯ ≥ β_m, σ(n) = ∑_{. d |n} d, and γ 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 _{m → ∞} ε _m(n) = 0.

Citations

• Approximate formulas for some functions of prime numbers
• An Elementary Problem Equivalent to the Riemann Hypothesis

Related

• Algorithm, probability, and prime numbers
• On the Riemann Hypothesis and the Difference Between Primes
• Probabilistic approach to the distribution of primes and to the proof of Legendre and Elliott-Halberstam conjectures
• Towards an Elementary Formulation of the Riemann Hypothesis in Terms of Permutation Groups
• Proof of the the Riemann hypothesis from the density and Lindelof hypotheses via a power sum method