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

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

2011-12-21

arxiv.org/abs/1112.4910v1

Mathematics

Number Theory

7 pages, 1 table; In "Herman J. J. te Riele Liber Amicorum", CWI, Amsterdam, December 2010

Scientific paper

We consider the real part Re(zeta(s)) of the Riemann zeta-function zeta(s) in the half-plane Re(s) >= 1. We show how to compute accurately the constant sigma_0 = 1.19... 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)) <= 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.

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