Strange and pseudo-differentiable functions with applications to prime partitions

TL;DR

This work studies asymptotics for partitions of $n$ into $r$-full primes $\mathfrak{p}_{\mathbb{P}_r}(n)$ and related weighted partitions $\mathfrak{p}_{\Lambda^{*r}}(n)$ using the Hardy-Littlewood circle method. It introduces the novel concept of strange/pseudo-differentiable functions to control non-principal major arcs, enabling precise main-term formulas expressed via polynomials $P_r$ and $\widetilde{P}_r$ of degree $r-1$ and a saddle-point analysis. The authors derive explicit asymptotics for both partition types, connect the results to zeros of the Riemann zeta-function, and provide a rigorous framework to handle higher-order arc contributions. The methodology extends prior prime-partition results, offering a robust toolkit (including contour integration and the saddle-point method) that can be adapted to other weighted prime-pattern partition problems and deeper connections to $\zeta$-zero phenomena.

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.

Figures (2)

• Figure 2.1: Domain plot of $\Phi_{\mathbb{P},M}(z)= \prod_{p \in \mathbb{P} \cap [1,M]} (\frac{1}{1-z^p})$ with $M=50$ (left) and $M=100$ (right) and absolute value and argument hue.
• Figure 8.1: The contour of integration $\Xi$ alongside the zero-free region of $\zeta(s)$.

Theorems & Definitions (68)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof : Proof of Lemma \ref{['lem:int_loglog_a_log_b']}
• Theorem 4.1: Generalized version of Vinogradov's lemma, DoRoZaZe24
• Theorem 4.2
• Lemma 4.3
• proof