Periodic vanishings of the Legendre-17 signed partition numbers

Abstract

For $f : \mathbb{N} \to \{0,\pm 1\}$ the $f$-signed partition numbers $\mathfrak{p}(n,f)$ are defined to be the weighted partition sums \[ \mathfrak{p}(n,f) = \sum_{\substack{x_{1}+\cdots+x_{k} = n \\ x_{1} \geq \cdots \geq x_{k} > 0 \\ k \geq 1}} f(x_{1})f(x_{2})\cdots f(x_{k}). \] For prime $p > 2$, let $(\frac{\cdot}{p})$ denote the Legendre symbol modulo $p$. The first half of this paper derives Rademacher-style series formulae for the quantities $\mathfrak{p}(n,\pm(\frac{\cdot}{p}))$ for $p < 24$ satisfying $p \equiv 1 \pmod{4}$ (that is, for $p=5,13,17$), and the extensions to general $p \equiv 1 \pmod{4}$ are made apparent in our derivations. In the second half of this paper, the series formulae for $\mathfrak{p}(n,\pm(\frac{\cdot}{17}))$, as well as various properties of Dedekind sums and their "character-twisted" analogues, are used to establish that these two quantities are identically zero on certain (mod $34$)-arithmetic progressions.