Computational Methods for the Riemann Zeta Function
Published in University of Michigan Deep Blue System, 2015
Abstract: This project focuses on exploring different computational methods of the Riemann Zeta function. The zeta function is formally defined as $\zeta(s)$ and the Riemann Hypothesis states that all non-trivial zeros of $\zeta(s)$ lie on the critical line where $\Re(s) = 1/2$. Modern research and advanced algorithms have not been able to disprove the Riemann Hypothesis. More recently, a variation of the Odlyzko-–Schönhage algorithm has verified the Riemann Hypothesis up to $10^{13}$ zeros. This project is composed of two major programs which calculate functional values and zeros of the Riemann zeta function. The first program calculates $\zeta(s)$ for any value of $s$ provided $s\in\mathbb{R}$ through what is known as the Cauchy-Schlömilch transformation. The second program uses the well known Riemann–Siegel formula. Prior to the work done inside the Odlyzko–Schönhage algorithm, this was the method of choice for finding zeros of $\zeta(s)$. The algorithm is heavily dependent on locating sign changes of the Riemann-Siegel $Z$ function in order to find zeros on the critical line. Loosely speaking, one can derive this formula from a direct relationship that was originally found by Siegel which states that $Z(t) = e^{i \theta(t)} \zeta\left(\frac{1}{2}+it\right)$.
Recommended citation: M. Kehoe (2015). “Computational methods for the Riemann zeta function.” MS Project. http://matthewshawnkehoe.github.io/files/kehoe_ms_project.pdf