On the Cesàro average of the "Linnik numbers"

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 $$\sum _{n\le N}{r}_Q\left(n\right)\frac{{\left(N-n\right)}^k}{\Gamma \left(k+1\right)}=M_1\left(N,k\right)+M_2\left(N,k\right)+M_3\left(N,k\right)+M_4\left(N,k\right)+O\left(N^{k+1}\right).$$ We also prove that with this technique the bound $k>3/2$ is optimal.

arXiv commentAccepted on Acta Arithmetica