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arXiv · 1402.0169

$a$-Points of the Riemann zeta-function on the critical line

S. J. Lester
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Abstract

We investigate the proportion of the nontrivial roots of the equation $ζ(s)=a$, which lie on the line $\Re s=1/2$ for $a \in \mathbb C$ not equal to zero. We show that at most one-half of these points lie on the line $\Re s=1/2$. Moreover, assuming a spacing condition on the ordinates of zeros of the Riemann zeta-function, we prove that zero percent of the nontrivial solutions to $ζ(s)=a$ lie on the line $\Re s=1/2$ for any nonzero complex number $a$.

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S. J. Lester. 2014-02-02. $a$-Points of the Riemann zeta-function on the critical line. https://doi.org/10.1093/imrn%2Frnt356

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