Collatz Cycles and $3n+c$ Cycles

TL;DR

This paper extends bounds for rational Collatz cycles to $3n+c$ cycles and introduces a new sequence with properties akin to Riemann zeta zeros, advancing understanding of these mathematical phenomena.

Contribution

It generalizes bounds for Collatz cycles to a broader class of $3n+c$ cycles and proposes a new sequence related to zeta function zeros.

Findings

Extended bounds for $3n+c$ cycles.

Introduced a sequence with properties similar to Riemann zeta zeros.

Enhanced understanding of cycle structures in number theory.

Abstract

Halbeisen and Hungerbuhler determined optimal bounds for the length of rational Collatz cycles. Their methods are extended to $3n+c$ cycles. Another sequence having properties similar to those of Riemann zeta function zeros is introduced.