Shan-Guang Tan | Searcharxiv

arXiv2025

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

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 \[ \lim_{x \to \infty}\frac{π(x+Φ(x))-π(x)}{Φ(x)/\log x}=1. \]

math.GM↗

arXiv2011

On the solution of the Collatz problem

In this paper, we first prove that given a nonnegative integer $m$ and an odd number $t$ not divisible by $3$, there exists a unique Collatz's Sequence \[ S_{c}(m,t)=\{n_{0}(m,t),n_{1}(m,t),n_{2}(m,t),\ldots,n_{m}(m,t),n_{m+1}(m,t)\} \] produced by a function $n_{i+1}(m,t)=(3n_{i}(m,t)+1)/2$ for $i=0,1,2,\ldots,m$ and ended by an even number $n_{m+1}(m,t)$ where $n_{i}(m,t)=2^{m+1-i}\times3^{i}t-1$ for $i=0,1,2,\ldots,m+1$, by which all odd numbers can be expressed. Then we discuss the Collatz problem in two ways and prove that each Collatz's Sequence always returns to 1, i.e., the Collatz problem is solved.

math.GM↗