Experimental Mathematics

On Repeated Values of the Riemann Zeta Function on the Critical Line

William D. Banks and Sarah Kang

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Let $\zeta (s)$ be the Riemann zeta function. In this paper, we study repeated values of $\zeta (s)$ on the critical line, and we give evidence to support our conjecture that for every nonzero complex number $z$, the equation $\zeta (1/2 + i t) = z$ has at most two solutions $t \in R$. We prove a number of related results, some of which are unconditional, and some of which depend on the truth of the Riemann hypothesis. We also propose some related conjectures that are implied by Montgomery’s pair correlation conjecture.

Article information

Experiment. Math. Volume 21, Issue 2 (2012), 132-140.

First available in Project Euclid: 31 May 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11M06: $\zeta (s)$ and $L(s, \chi)$

Riemann zeta function critical line repeated values loops


Banks, William D.; Kang, Sarah. On Repeated Values of the Riemann Zeta Function on the Critical Line. Experiment. Math. 21 (2012), no. 2, 132--140.https://projecteuclid.org/euclid.em/1338430826

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