# On the Ces\`aro average of the numbers that can be written as sum of a prime and two squares of primes

## TL;DR

This paper derives an asymptotic formula for the sum of the von Mangoldt-weighted count of numbers that are sums of a prime and two prime squares, extending understanding of their distribution.

## Contribution

It provides a new asymptotic expansion for the weighted count of numbers expressible as a prime plus two prime squares, involving specific parameters and error bounds.

## Key findings

- Asymptotic formula for the sum involving r_{SP}(n) and (N-n)^k
- Identification of main terms M_i(N,k) in the expansion
- Error term bounded by O(N^{k+1})

## 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 $$\sum_{n\leq N}r_{SP}\left(n\right)\frac{\left(N-n\right)^{k}}{\Gamma\left(k+1\right)}=M_{1}\left(N,k\right)+M_{2}\left(N,k\right)+M_{3}\left(N,k\right)+M_{4}\left(N,k\right)+O\left(N^{k+1}\right).$$