Exponential sums with polynomials and their applications to primes in sparse sets

TL;DR

This paper establishes new bounds for exponential sums with polynomials and applies these results to improve the understanding of prime distribution within sparse Piatetski-Shapiro sequences, including iterated sequences.

Contribution

It introduces a new upper bound for exponential sums with polynomials and enhances the asymptotic formulas for primes in Piatetski-Shapiro sequences and their iterates.

Findings

Improved upper bounds for exponential sums with polynomials.

Extended the admissible range for primes in Piatetski-Shapiro sequences.

Derived asymptotic formulas for primes in iterated Piatetski-Shapiro sequences.

Abstract

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an application, we study the Piatetski-Shapiro sequence of the form $(\lfloor n^c \rfloor)$ where $c > 1$ is not an integer. We improve the admissible range of the asymptotic formula for primes in the intersection of Piatetski-Shapiro sequences. We also study the iterated Piatetski-Shapiro sequence and prove an asymptotic formula for the prime counting function.