On $|{\rm Li}(x)-π(x)|$ and primes in short intervals
Shan-Guang Tan
TL;DR
The paper proves that $\\pi(x)=\\mathrm{Li}(x)+O(\\sqrt{x}\log x)$ by introducing x-dependent truncations $\\pi^{*}(x,N)$ to approximate both $\\mathrm{Li}(x)$ and $\\pi(x)$, and showing successive truncation steps have gaps bounded by $x^{1/64}$. It then derives a prime-number theorem in short intervals: for $\\Phi(x)=\\beta x^{1/2}$ (with a specific cutoff $x_{\\beta}$), we have $\\frac{\\pi(x+\\Phi(x))-\\pi(x)}{\\Phi(x)/\\log x}=1+O(1/\\log x)$ and the limit equals 1. The core technique is making the truncation level depend on $x$ and analyzing discriminants via $\\alpha_{L,M}$ and $\\rho_{L}(x,M)$ to tightly bound error terms. This yields a unified framework for both global prime distribution (through Li–pi closeness) and primes in short intervals without assuming the Riemann Hypothesis. Overall, the work provides quantitative, region-wise control of prime counts and sharp asymptotics for primes in short ranges.
Abstract
Two topics of the number theory are discussed in this paper. First, we prove that given each natural number $x\geq10^{3}$, we have \[ |{\rm Li}(x)-π(x)|\leq c\sqrt{x}\log x\texttt{ and } π(x)={\rm Li}(x)+O(\sqrt{x}\log x) \] 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 $x\geq10^{41}$, 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 $x\geq x_β$ where $Φ(x):=x^{1/2}$ when let $β=1$. Then there are \[ \frac{π(x+Φ(x))-π(x)}{Φ(x)/\log x}=1+O(\frac{1}{\log x}) \] and \[ \lim_{x \to \infty}\frac{π(x+Φ(x))-π(x)}{Φ(x)/\log x}=1. \]