Legendre-signed partition numbers

Let $f:\mathbb{N}\to \{0,\pm 1\}$, for $n\in \mathbb{N}$ let $\mathrm{\Pi }[n]$ be the set of partitions of $n$, and for all partitions $\pi =(a_1,a_2,\dots ,a_k)\in \mathrm{\Pi }[n]$ let $$f(\pi ):=f(a_1)f(a_2)\cdots f(a_k)\mathrm{.}$$ With this we define the $f$-signed partition numbers $$\mathfrak{p}(n,f)=\sum _{\pi \in \mathrm{\Pi }[n]}f(\pi )\mathrm{.}$$ In this paper, for odd primes $p$ we derive asymptotic formulae for $\mathfrak{p}(n,{\chi }_p)$ as $n\to \mathrm{\infty }$, 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 $j\ge 0$ 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 $p\mathrm{\ne }5$ such that $p\equiv ̸1(\mathrm{m}\mathrm{o}\mathrm{d}8)$. 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 $\mathrm{\mid }f\mathrm{\mid }\le 1$.