On |\rm Li(x)-π(x)| and primes in short intervals

2011/10/12 by Shan-Guang Tan, Tan, Shan-Guang
Mathematics · #11A41 #11M26 #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM)

paper · pdf · doi:10.48550/arxiv.1110.2952

openalex publication_date 2011/10/12 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/31

Abstract

Two topics of the number theory are discussed in this paper. First, we prove that given each natural number x≥10³, we have |\rm Li(x)-π(x)|≤ c√(x)log x and π(x)=\rm Li(x)+O(√(x)log x) where c is a constant greater than 1 and less than e. Second, with a much more accurate estimation of prime numbers, the error range of which is less than x^{1/2-0.0327283} for x≥10⁴¹, we prove a theorem of the number of primes in short intervals: Given a positive real number β that determines a real number x_β by e(log x_β)³/x_β^0.0327283=β, let Φ(x):=βx^1/2 for x≥ x_β where Φ(x):=x^1/2 when let β=1. Then there are (π(x+Φ(x))-π(x))/(Φ(x)/log x)=1+O((1)/(log x)) and lim_{x → ∞}(π(x+Φ(x))-π(x))/(Φ(x)/log x)=1.

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