Riemann Zeta Search Project

Prime numbers are the building blocks of mathematics and affects every part of life. It is well known that there are infinitely many primes, as proved by Euclid around 300 BC. In the 18th century Gauss conjectured that the number of primes up to x is asymptotically $x/\ln{x}$ . Many new results has been achieved in the past century with respect to the prime numbers, but the distribution of primes remained unknown. The Riemann zeta function was introduced and studied by Leonhard Euler in the first half of the eighteenth century without using complex analysis. Let $s=\sigma+it$ where $\sigma , t \in \mathbb{R}$ . The Riemann zeta function is defined by

$\begin{equation*}\zeta(s)=\sum\limits_{n=1}^\infty n^{-s} \qquad (\sigma>1)\end{equation*}$

which is an extremely important function of mathematics and physics. In 1859 Bernhard Riemann conjectured that all nontrivial zeros of the Riemann zeta function have real part $\sigma=1/2$ . This is the famous Riemann-hypothesis, one of the most important unsolved problem in the theory of prime numbers. The Riemann zeta function can be calculated on the critical line by using the Riemann-Siegel $Z$ -function. The function $Z(t)$ can be calculated in time complexity of $\mathcal{O}(t^{1/2})$ by

$\begin{equation*}Z(t)=2\sum\limits_{n=1}^{\lfloor\sqrt{t/2\pi}\rfloor} \frac{1}{\sqrt{n}}\cos(\theta(t)-t\cdot\ln{n})+\mathcal{O}(t^{-1/4})\,\end{equation*}$

where $\theta(t)=\arg(\Gamma(1/4+\frac{it}{2}))-\frac{1}{2}t\ln{\pi}$ .In 2011 Ghaith A. Hiary presented how to compute $Z(t)$ within $\mathcal{O}(t^{2/5}), \mathcal{O}(t^{1/3})$ and $\mathcal{O}(t^{4/{13}})$ time complexities, respectively. Since $\zeta(1/2+it)$ is unbounded, $Z(t)$ can take arbitrarily large values as $t$ goes to infinity.

In 2013 a method was published for solving special types of $n$ -dimensional Diophantine approximation problems efficiently. Based on this result a new algorithm RS-Peak was presented for locating peak values of the Riemann zeta function on the critical line. Applying RS-Peak on the SZTAKI Desktop Grid of the Hungarian Academy of Science thousands of large $|Z(t)|>10.000$ values found.

The aim of the Riemann Zeta Search Project is locating peak values of the zeta function on the critical line in order to have a better understanding of the distribution of prime numbers.

On 30th of November, 2015 the Riemann Zeta Search Project found the largest known $|Z(t)|>10.000$ . For $t = 310678833629083965667540576593682.05$ we have $Z(t) \approx$ $16874.202$ .

Riemann Zeta Search Project

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Large Z(t) values

Riemann Zeta Search Project