A ternary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$

TL;DR

This paper proves that for large numbers, the ternary Piatetski-Shapiro inequality can be solved with primes of a specific form, using a new exponential sum estimate over primes.

Contribution

It introduces a novel Bombieri-Vinogradov type theorem for exponential sums over primes of a special form, enabling solutions to the inequality.

Findings

Solutions exist for large N with primes of the form p=x^2+y^2+1

Established a new Bombieri-Vinogradov type result for exponential sums

Extended the range of c for which the inequality holds

Abstract

In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $1<c<\frac{427}{400}$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$, such that $p_1=x^2 + y^2 +1$. For this purpose we establish a new Bombieri -- Vinogradov type result for exponential sums over primes.