On the Ces\`aro average of the "Linnik numbers"

TL;DR

This paper studies the Cesàro average of Linnik numbers, which are integers expressible as a prime plus two squares, providing an asymptotic formula involving Bessel functions and establishing the optimality of the parameter range.

Contribution

It introduces a new asymptotic formula for the Cesàro average of Linnik numbers using Bessel functions and proves the optimality of the parameter condition k>3/2.

Findings

Derived an asymptotic expression involving multiple terms M_i(N,k)

Established the bound k>3/2 as optimal for the method

Provided error estimates of order N^{k+1}

Abstract

Let $\Lambda$ be the von Mangoldt function and $r_{Q}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}\Lambda\left(m_{1}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares (that we will call Linnik numbers, for brevity). Let $N$ a sufficiently large integer, let $k>3/2$ and let $M_{i}\left(N,k\right),\, i=1,\dots,k$ suitable parameters depending on $J_{v}\left(u\right)$, where $J_{v}\left(u\right)$ denotes the Bessel function of complex order $v$ and real argument $u$. We prove that $$ \sum_{n\leq N}r_{Q}\left(n\right)\frac{\left(N-n\right)^{k}}{\Gamma\left(k+1\right)}=M_{1}\left(N,k\right)+M_{2}\left(N,k\right)+M_{3}\left(N,k\right)+M_{4}\left(N,k\right)+O\left(N^{k+1}\right).$$ We also prove that with this technique the bound $k>3/2$ is optimal.