Biasymptotics of the M\"obius- 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 $(x_1,x_2,\ldots,x_k)$ such that $x_1 \geq x_2 \geq \cdots \geq x_k$ and $x_1 + x_2 + \cdots + x_k = n$. Fixing $f:\mathbb{N}\to\{0,\pm 1\}$, for $\pi = (x_1,x_2,\ldots,x_k) \in \Pi[n]$ let $f(\pi) := f(x_1)f(x_2)\cdots f(x_k)$. In this way we define the {signed partition numbers} \[ p(n,f) = \sum_{\pi\in\Pi[n]} f(\pi). \] Building on the author's previous work on the quantities $p(n,\mu)$ and $p(n,\lambda)$, where $\mu$ and $\lambda$ are the M\"obius 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.