Department of Mathematics and Statistics

Number Theory

Zeros of the derivatives of the Riemann zeta function

The plot shows zeros k of the derivatives $\zeta^{(k)}(\sigma+it)$ of the Riemann Zeta functionon the complex plane. In 1965 Spira had already noticed that the zeros of $\zeta'(s)$ and $\zeta''(s)$ seem to come in pairs, where the zero of $\zeta''(s)$ is always located to the right of the zero of $\zeta'(s)$. With the help of extensive computations, Skorokhodov (2003) observed this behavior for higher derivatives as well. For large $k$ and $\sigma$ this phenomenon is proven in New Zero-Free Regions for the Derivatives of the Riemann Zeta Function by Thomas Binder, Sebastian Pauli, and Filip Saidak.