Sum of one prime and two squares of primes in short intervals

TL;DR

Under the assumption of the Riemann Hypothesis, the paper proves that sufficiently large short intervals contain integers that are sums of one prime and two squares of primes, with the interval length depending on a logarithmic power.

Contribution

The paper establishes a conditional result on the distribution of numbers representable as a prime plus two prime squares within short intervals, assuming the Riemann Hypothesis.

Findings

Interval length $H \

contains such numbers under the Riemann Hypothesis

Provides an explicit bound on $H$ in terms of $(\log N)^{4}$

Abstract

Assuming the Riemann Hypothesis we prove that the interval $[N, N + H]$ contains an integer which is a sum of a prime and two squares of primes provided that $H \ge C (\log N)^{4}$, where $C > 0$ is an effective constant.