Strange and pseudo-differentiable functions with applications to prime partitions

TL;DR

This paper develops new methods involving strange functions and pseudo-differentiability to analyze prime partitions and their asymptotics, extending classical results and connecting to the zeros of the Riemann zeta-function.

Contribution

It introduces the concepts of strange functions and pseudo-differentiability to study prime partitions and related convolutions, providing new asymptotic formulas and analytic techniques.

Findings

Asymptotic formula for prime partitions into r-full primes

Extension of classical prime partition results

Connection to zeros of the Riemann zeta-function

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.