On the Cesàro average of the “Linnik numbers”

Marco Cantarini¹

¹ Università di Ferrara Dipartimento di Matematica e Informatica Via Machiavelli, 30 44121 Ferrara, Italy

Acta Arithmetica, Tome 180 (2017) no. 1, pp. 45-62.

Résumé

Let $\varLambda$ be the von Mangoldt function and $$ r_{Q}(n)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}\varLambda(m_{1}) $$ 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 $N$ be a sufficiently large integer. We prove that for $k \gt 3/2$ we have $$ \sum_{n\leq N}r_{Q}(n)\frac{(N-n)^{k}}{\varGamma(k+1)}=M(N,k)+O(N^{k+1}) $$ where $M(N,k)$ is essentially a weighted sum, over non-trivial zeros of the Riemann zeta function, of Bessel functions of complex order and real argument. We also prove that with this technique the bound $k \gt 3/2$ is optimal.

Détail

DOI : 10.4064/aa8601-3-2017

Keywords: varlambda von mangoldt function sum varlambda counting function numbers written sum prime squares call linnik numbers brevity sufficiently large integer prove have sum leq frac n n vargamma where essentially weighted sum non trivial zeros riemann zeta function bessel functions complex order real argument prove technique bound optimal