Biasymptotics of the Möbius- and Liouville-signed partition numbers

Abstract

For $n\in \mathbb{N}$ let $\Pi [n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\dots ,x_k)$ such that $x_1\ge x_2\ge \cdots \ge x_k$ and $x_1+x_2+\cdots +x_k=n$. Fixing $f:\mathbb{N}\to \{0,\pm 1\}$, for $\pi =(x_1,x_2,\dots ,x_k)\in \Pi [n]$ let $f(\pi ):=f(x_1)f(x_2)\cdots f(x_k)$. In this way we define the {signed partition numbers} \[ p(n,f) = \sum_{π\inΠ[n]} f(π). \] Building on the author's previous work on the quantities $p(n,\mu )$ and $p(n,\lambda )$, where $\mu$ and $\lambda$ are the Möbius and Liouville functions of prime number theory, respectively, on assumptions about the zeros of the Riemann zeta function we establish an alternation of the terms $p(n,\mu )$ between two asymptotic behaviors as $n\to \infty$. Similar results for the quantities $p(n,\lambda )$ are established. However, it is also demonstrated that if the Riemann Hypothesis (RH) holds, then it is possible that the quantities $p(n,\lambda )$ maintain a single asymptotic behavior as $n\to \infty$. In particular, this stable asymptotic behavior occurs if, in addition to RH, it holds that all zeros of $\zeta (s)$ in the critical strip $\{0<\Re (s)<1\}$ are simple and the residues of $1/\zeta (s)$ at these zeros are not too large. To formally describe these stable and alternating behaviors, the notions of asymptotic and biasymptotic sequences are introduced using a modification of the real logarithm.

38 pages, 4 figures