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A proof of the Riemann hypothesis based on the Koch theorem, on primes in short intervals, and distribution of nontrivial zeros of the Riemann zeta function
| Content Provider | Semantic Scholar |
|---|---|
| Author | Tan, Shan-Guang |
| Copyright Year | 2011 |
| Abstract | Part One: How to get more accurate estimation of prime numbers is an important problem in number theory. In this paper, we obtain much more accurate estimation of prime numbers, the error range of which is less than $\sqrt{x\log x}$ for $x\geq10^{3}$ or less than $\sqrt{x\log x}/x^{0.0327283}$ for $x\geq10^{41}$. These results shall be important and useful for researches on Riemann hypothesis and on primes in short intervals. Part Two: In 1901, H. Koch showed that if and only if the Riemann hypothesis is true, then $\pi(x)=\Li(x)+O(\sqrt{x}\log x)$. Let define \[ \eta^{*}(x,N)=\sum_{n=0}^{N}\frac{n!}{\log^{n}x}\texttt{ and } \pi^{*}(x,N)=\frac{x}{\log x}\eta^{*}(x,N)=\frac{x}{\log x}\sum_{n=0}^{N}\frac{n!}{\log^{n}x} \] where we have proved that the pair of numbers $x$ and $N$ in $\pi^{*}(x,N)$ satisfy inequalities $\pi^{*}(x,N)<\pi(x)<\pi^{*}(x,N+1)$, and the number $N$ is a non-decreasing step function of the variable $\log x$ for $x\geq10^{3}$ and approximately proportional to $\log x$. Then we write \[ |\pi(x)-\Li(x)|\leq|\pi(x)-\pi^{*}(x,N)|+|\Li(x)-\pi^{*}(x,N)|. \] In an early paper, we have proved $\pi(x)-\pi^{*}(x,N)<\sqrt{x\log x}$. In this paper, we prove the estimation $\Li(x)=\pi^{*}(x,N)+O(\sqrt{x\log x})$. So we obtain $\pi(x)=\Li(x)+O(\sqrt{x\log x})$. Hence the Riemann hypothesis is true. Part Three: We prove a theorem: Let $\Phi(x)=\beta x^{1/2}$, $0<\beta\leq1$. For $x\geq e^{470}$ such that $(\log x)^{5/2}\leq x^{0.0327283}$ there are \[ \frac{\pi(x+\Phi(x))-\pi(x)}{\Phi(x)/\log x}=1+O(\frac{1}{\log x}) \] and \[ \lim_{x \to \infty}\frac{\pi(x+\Phi(x))-\pi(x)}{\Phi(x)/\log x}=1. \] Part Four: Based on the results above, some famous conjectures of distribution of primes in short intervals, such as Legendre's conjecture, Oppermann's conjecture, Brocard's conjecture and Andrica's conjecture, can be proved. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/1110.2952v20.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
