Strange and pseudo-differentiable functions with applications to prime partitions

Abstract

Let ${\mathfrak{p}}_{{\mathbb{P}}_r}(n)$ denote the number of partitions of $n$ into $r$-full primes. We use the Hardy-Littlewood circle method to find the asymptotic of ${\mathfrak{p}}_{{\mathbb{P}}_r}(n)$ as $n\to \infty$. This extends previous results in the literature of partitions into primes. We also show an analogue result involving convolutions of von Mangoldt functions and the zeros of the Riemann zeta-function. To handle the resulting non-principal major arcs we introduce the definition of strange functions and pseudo-differentiability.

Pages: 50. Figures: 7. Keywords: weights associated to partitions, pseudo-differentiable functions, strange functions, inclusion-exclusion, Hardy-Littlewood circle method, exponential sums, von Mangoldt function, zeros of the zeta function. Updates: Section 5 has been split into several sections and slightly reorganised, and some minor typos have been corrected