Corrigendum for "The generalized strong recurrence for non-zero rational parameters" Archiv der Mathematik 95 (2010), 549-555

TL;DR

This paper establishes equivalences between the self-approximation of the logarithm of the Riemann zeta function and the Riemann Hypothesis, extending previous results and filling gaps related to rational parameters.

Contribution

It proves the equivalence of self-approximation of log zeta with the Riemann Hypothesis and extends self-approximation results to all nonzero real numbers d, filling gaps in prior work.

Findings

Self-approximation of log ζ(s) with d=0 is equivalent to the Riemann Hypothesis.

Self-approximation of log ζ(s) holds for all nonzero real d.

Self-approximation of ζ(s) for rational d with |a-b|≠1 and gcd(a,b)=1 is established.

Abstract

In the present paper, we prove that self-approximation of $\log \zeta (s)$ with $d=0$ is equivalent to the Riemann Hypothesis. Next, we show self-approximation of $\log \zeta (s)$ with respect to all nonzero real numbers $d$. Moreover, we partially filled a gap existing in "The strong recurrence for non-zero rational parameters" and prove self-approximation of $\zeta(s)$ for $0 \ne d=a/b$ with $|a-b|\ne 1$ and $\gcd(a,b)=1$.