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On the Cesàro average of the “Linnik numbers”

Volume 180 / 2017

Marco Cantarini Acta Arithmetica 180 (2017), 45-62 MSC: Primary 11P32; Secondary 44A10, 33C10. DOI: 10.4064/aa8601-3-2017 Published online: 9 August 2017

Abstract

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.

Authors

  • Marco CantariniUniversità di Ferrara
    Dipartimento di Matematica e Informatica
    Via Machiavelli, 30
    44121 Ferrara, Italy
    e-mail

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