Proof of the Linderlof hypothesis

Details Proof of the Linderlof hypothesis Proof of the Linderlof hypothesis

2010-10-16

arxiv.org/abs/1010.3374v1

Mathematics

General Mathematics

Scientific paper

The Riemann zeta function is a meromorphic function on the whole complex plane. It has infinitely many zeros and a unique pole at $s = 1$. Those zeros at $s = -2, -4, -6, ...$ are known as trivial zeros. The Riemann hypothesis is a conjecture of Riemann, from 1859, claiming that all nontrivial zeros of $\zeta(s)$ lie on the line $\Re(s) =\tfrac{1}{2}$. Let $x \ge2$. Define $\Lambda(n) =\log p$ whenever $n =p\sp{m}$ for a prime number $p$ and a positive integer $m$, and zero otherwise. The Riemann hypothesis is then equivalent to the $\psi$-form of the prime number theorem as $\psi(x) -x =O(x\sp{1/2} \log\sp{2} x)$, where $\psi(x) =\sum\sb{n\le x}\ \Lambda(n)$ with the sum running through the set of all natual integers. Similar to that in the literature, we use the classical integral formula for the Heaviside function in the form of $H(x) =\int\sb{m -i\infty} \sp{m +i\infty} \tfrac{x\sp{s}}{s} \dd s$ where $m >0$, and $H(x)$ is 0 when $x <1$, $\tfrac{1}{2}$ when $x=1$, and 1 when $x >1$. Starting with the estimate on $\psi(x) - x =O(x \log x)$, we use induction to reduce the size of the exponent $\theta=\theta(x)$ in $\varpi(x) =O(x\sp{\theta} \log\sp{2} x)$, while we also use induction on $x$ when $\theta(x)$ is fixed. We prove the Riemann hypothesis.

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