An Analytic Approach to the Collatz $3n+1$ Problem for Negative Start Values

Abstract

In three papers, Meinardus (Über das Syracuse-Problem, Preprint Nr. 67, Universität Mannheim, 1987) and Berg and Meinardus (Results Math 25:1–12, 1994; Rostock Math Kolloq 48:11–18, 1995) have shown that the Collatz $3n+1$ problem for positive integers $n$ as start values can be put into the theory of complex analysis. Here we investigate the Collatz $3n+1$ problem for negative start values. This problem is equivalent to the $3n-1$ problem for positive start values. It is known, that this problem differs from the $3n+1$ problem. One aspect is, that all positive start values tend, at least empirically, to either 1, 5 or 17. We describe the corresponding analytic problem for this case, where one has to show that there are not more than three linearly independent, holomorphic solutions for this problem. We conjecture that these solutions have the unit circle as natural boundary. However, the $3n-1$ problem remains open.