On the Cesàro average of the numbers that can be written as sum of a prime and two squares of primes

Abstract

Let $\Lambda \left(n\right)$ be the von Mangoldt function and $r_{SP}\left(n\right)={\sum }_{m_1+m_2^2+m_3^2=n}\Lambda \left(m_1\right)\Lambda \left(m_2\right)\Lambda \left(m_3\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares of primes. Let $N$ a sufficiently large integer, let $k>3/2$ and let $M_i\left(N,k\right),i=1,\dots ,4$ suitable parameters depending on $\Gamma (s)$. We prove that $$\sum _{n\le N}{r}_{SP}\left(n\right)\frac{{\left(N-n\right)}^k}{\Gamma \left(k+1\right)}=M_1\left(N,k\right)+M_2\left(N,k\right)+M_3\left(N,k\right)+M_4\left(N,k\right)+O\left(N^{k+1}\right).$$

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