Strange and pseudo-differentiable functions with applications to prime partitions

Dong, Anji and Robles, Nicolas and Zaharescu, Alexandru and Zeindler, Dirk (2025) Strange and pseudo-differentiable functions with applications to prime partitions. Research in Number Theory, 11 (2): 52. ISSN 2522-0160

Abstract

Let pPr(n) denote the number of partitions of n into r-full primes. We use the Hardy–Littlewood circle method to find the asymptotic of pPr(n) as n→∞. 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.