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

2025/10/23 by Lingyu Guo, Guo, Lingyu, Victor Zhenyu Guo +3
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2510.20562

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.

Citations

• Decoupling, exponential sums and the Riemann zeta function
• Piatetski-Shapiro sequences
• Nombres premiers de la forme ⌊nc⌋
• The Pjateckiĭ-S̆apiro prime number theorem
• Prime Numbers in Short Intervals and a Generalized Vaughan Identity

Related

• Improvements on exponential sums related to Piatetski-Shapiro primes
• The Piatetski-Shapiro prime number theorem
• Exponential sums over Piatetski-Shapiro primes in arithmetic progressions
• Exponential sums with sparse polynomials over finite fields
• Multilinear Exponential Sums With A General Class Of Weights