## Illinois Journal of Mathematics

### Zeros of $\zeta^{\prime} (s)$ and the Riemann hypothesis

Robert Spira

#### Abstract

It is shown that the Riemann hypothesis implies that the derivative of the Riemann zeta function has no zeros in the open left half of the critical strip. It is also shown, with no hypothesis, that, with the exception of a bounded region where the zeros can be calculated, the closed left half plane contains only real zeros of the derivative. It is further shown that the Riemann hypothesis is equivalent to the condition that $|\zeta(s)|$ increases as $\mathrm{Re}\,s$ moves left from $1/2$ for $\mathrm{Im}\,s$ sufficiently large.

#### Article information

Source
Illinois J. Math. Volume 17, Issue 1 (1973), 147-152.

Dates
First available in Project Euclid: 20 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1256052045

Mathematical Reviews number (MathSciNet)
MR0309881

Zentralblatt MATH identifier
0247.10022

Subjects
Primary: 10H05

#### Citation

Spira, Robert. Zeros of $\zeta^{\prime} (s)$ and the Riemann hypothesis. Illinois J. Math. 17 (1973), no. 1, 147--152. https://projecteuclid.org/euclid.ijm/1256052045