Legendre-signed partition numbers

Abstract

Let $f:\mathbb{N}\to\{0,\pm 1\}$, for $n \in \mathbb{N}$ let $\Pi[n]$ be the set of partitions of $n$, and for all partitions $\pi = (a_1,a_2,\ldots,a_k) \in \Pi[n]$ let \[ f(\pi) := f(a_1)f(a_2) \cdots f(a_k). \] With this we define the $f$-signed partition numbers \[ \mathfrak{p}(n,f) = \sum_{\pi\in\Pi[n]} f(\pi). \] In this paper, for odd primes $p$ we derive asymptotic formulae for $\mathfrak{p}(n,\chi_p)$ as $n\to\infty$, where $\chi_p(n)$ is the Legendre symbol $(\frac{n}{p})$ associated $p$. A similar asymptotic formula for $\mathfrak{p}(n,\chi_2)$ is also established, where $\chi_2(n)$ is the Kronecker symbol $(\frac{n}{2})$. Special attention is paid to the sequence $(\mathfrak{p}(n,\chi_5))_\mathbb{N}$, and a formula for $\mathfrak{p}(n,\chi_5)$ supporting the recent discovery that $\mathfrak{p}(10j+2,\chi_5)=0$ for all $j\geq 0$ is discussed. Our main results imply, as a corollary, that the periodic vanishing displayed by $(\mathfrak{p}(n,\chi_5))_\mathbb{N}$ does not occur in any sequence $(\mathfrak{p}(n,\chi_p))_\mathbb{N}$ for $p \neq 5$ such that $p\not\equiv 1\,\,(\mathrm{mod}\,8)$. In addition, work of Montgomery and Vaughan on exponential sums with multiplicative coefficients is applied to establish an upper bound on certain doubly infinite series involving multiplicative functions $f$ with $|f| \leq 1$.