A Ces\`aro Average of Hardy-Littlewood numbers

TL;DR

This paper derives an explicit Cesàro average formula for Hardy-Littlewood numbers involving the von Mangoldt function, Riemann zeta zeros, and Bessel functions, revealing deep connections between prime representations and special functions.

Contribution

It provides a new explicit formula for the Cesàro average of Hardy-Littlewood numbers, incorporating zeros of the Riemann zeta-function and Bessel functions, extending previous results in additive number theory.

Findings

Explicit Cesàro average formula involving zeta zeros and Bessel functions

Connections between prime representations and special functions

Asymptotic behavior characterized for large N

Abstract

Let $\Lambda$ be the von Mangoldt function and $r_{\textit{HL}}(n) = \sum_{m_1 + m_2^2 = n} \Lambda(m_1),$ be the counting function for the Hardy-Littlewood numbers. Let $N$ be a sufficiently large integer. We prove that $$\begin{align}\sum_{n \le N} r_{\textit{HL}}(n) \frac{(1 - n/N)^k}{\Gamma(k + 1)} &= \frac{\pi^{1 / 2}}2 \frac{N^{3 / 2}}{\Gamma(k + 5 / 2)} - \frac 12 \frac{N}{\Gamma(k + 2)} - \frac{\pi^{1 / 2}}2 \sum_{\rho} \frac{\Gamma(\rho)}{\Gamma(k + 3 / 2 + \rho)} N^{1 / 2 + \rho}\\ &+ 1/2 \sum_{\rho} \frac{\Gamma(\rho)}{\Gamma(k + 1 + \rho)} N^{\rho} + \frac{N^{3 / 4 - k / 2}}{\pi^{k + 1}} \sum_{\ell \ge 1} \frac{J_{k + 3 / 2} (2 \pi \ell N^{1 / 2})}{\ell^{k + 3 / 2}}\\ &- \frac{N^{1 / 4 - k / 2}}{\pi^k} \sum_{\rho} \Gamma(\rho) \frac{N^{\rho / 2}}{\pi^\rho} \sum_{\ell \ge 1} \frac{J_{k + 1 / 2 + \rho} (2 \pi \ell N^{1 / 2})} {\ell^{k + 1 / 2 + \rho}} + \mathcal{O}_k(1).\end{align}$$ for $k > 1$, where $\rho$ runs over the non-trivial zeros of the Riemann zeta-function $\zeta(s)$ and $J_{\nu} (u)$ denotes the Bessel function of complex order $\nu$ and real argument $u$.