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arXiv · 2412.20102

Strange and pseudo-differentiable functions with applications to prime partitions

Anji Dong·Nicolas Robles·Alexandru Zaharescu·Dirk Zeindler
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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.

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Anji Dong, Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler. 2024-12-28. Strange and pseudo-differentiable functions with applications to prime partitions. https://doi.org/10.1007/s40993-025-00628-8

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