Sums of one prime power and two squares of primes in short intervals

Abstract

Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_1^k+p_2^2+p_3^2$, where $p_1$, $p_2$, $p_3$ are prime numbers, holds in intervals shorter than the ones previously known.