Legendre-signed partition numbers

Abstract

Let $f:\mathbb{N}\to \{0,\pm 1\}$, for $n\in \mathbb{N}$ let $\hat{I}[n]$ be the set of partitions of $n$, and for all partitions $\ddot{I}€=(a_1,a_2,\dots ,a_k)\in \hat{I}[n]$ let \[ f(π) := f(a_1)f(a_2) \cdots f(a_k). \] With this we define the $f$-signed partition numbers \[ \mathfrak{p}(n,f) = \sum_{π\inÎ [n]} f(π). \] In this paper, for odd primes $p$ we derive asymptotic formulae for $\mathfrak{p}(n,\ddot{I}{‡}_p)$ as $n\to \infty$, where $\ddot{I}{‡}_p(n)$ is the Legendre symbol $(\frac{n}{p})$ associated $p$. A similar asymptotic formula for $\mathfrak{p}(n,\ddot{I}{‡}_2)$ is also established, where $\ddot{I}{‡}_2(n)$ is the Kronecker symbol $(\frac{n}{2})$. Special attention is paid to the sequence $(\mathfrak{p}(n,\ddot{I}{‡}_5){)}_{\mathbb{N}}$, and a formula for $\mathfrak{p}(n,\ddot{I}{‡}_5)$ supporting the recent discovery that $\mathfrak{p}(10j+2,\ddot{I}{‡}_5)=0$ for all $j\ge 0$ is discussed. Our main results imply, as a corollary, that the periodic vanishing displayed by $(\mathfrak{p}(n,\ddot{I}{‡}_5){)}_{\mathbb{N}}$ does not occur in any sequence $(\mathfrak{p}(n,\ddot{I}{‡}_p){)}_{\mathbb{N}}$ for $p\ne 5$ 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 $|f|\le 1$.

Updated version with changes made during publication. 39 pages, 2 figures