Large sieve estimate for multivariate polynomial moduli and applications

TL;DR

This paper establishes advanced large sieve inequalities for multivariate polynomial moduli, leading to a Bombieri--Vinogradov type theorem that improves prime distribution estimates related to polynomial values.

Contribution

It introduces new large sieve bounds for multivariate polynomial moduli and extends prime distribution results to a broader class of polynomials with more variables.

Findings

Proves large sieve inequalities for multivariate polynomial moduli.

Derives a Bombieri--Vinogradov type theorem for polynomial moduli.

Shows existence of infinitely many primes with large prime divisors related to polynomial values.

Abstract

We prove large sieve inequalities with multivariate polynomial moduli and deduce a general Bombieri--Vinogradov type theorem for a class of polynomial moduli having a sufficient number of variables compared to its degree. This sharpens previous results of the first author in two aspects: the range of the moduli as well as the class of polynomials which can be handled. As a consequence, we deduce that there exist infinitely many primes $p$such that $p-1$ has a prime divisor of size $\gg p^{2/5+o(1)}$ that is the value of an incomplete norm form polynomial.