A note on the real part of the Riemann zeta-function

Abstract

We consider the real part $\Re (\zeta (s))$ of the Riemann zeta-function $\zeta (s)$ in the half-plane $\Re (s)\ge 1$. We show how to compute accurately the constant ${\sigma }_0=1.19\dots$ which is defined to be the supremum of $\sigma$ such that $\Re (\zeta (\sigma +it))$ can be negative (or zero) for some real $t$. We also consider intervals where $\Re (\zeta (1+it))\le 0$ and show that they are rare. The first occurs for $t$ approximately 682112.9, and has length about 0.05. We list the first fifty such intervals.

8 pages, 1 table; In "Herman J. J. te Riele Liber Amicorum", CWI, Amsterdam, December 2010. Eqn. (4) corrected in v2

References in corpus (1)

• A method for locating where the real part of the Riemann zeta function becomes negative for its real argument greater than one

Cited by in corpus (2)

• On the sign of the real part of the Riemann zeta-function
• Errata and Addenda to Mathematical Constants