Weighted exponential sums and its applications

TL;DR

This paper investigates the distribution of polynomial sequences modulo one for specific classes of integers and evaluates weighted exponential sums involving divisor and Möbius functions, providing non-trivial estimates in minor arc regions.

Contribution

It introduces new results on the distribution of polynomial values modulo one for integers with certain divisor properties and provides bounds for exponential sums weighted by divisor and Möbius functions.

Findings

Distribution results for polynomial sequences modulo one for integers with at least three divisors.

Non-trivial estimates for weighted exponential sums involving divisor and Möbius functions.

Analysis of exponential sums when the polynomial's leading coefficient is in the minor arc.

Abstract

Let $f$ be a real polynomial with irrational leading co-efficient. In this article, we derive distribution of $f(n)$ modulo one for all $n$ with at least three divisors and also we study distribution of $f(n)$ for all square-free $n$ with at least two prime factors. We study exponential sums when weighted by divisor functions and exponential sums over square free numbers. In particular, we are interested in evaluating \begin{align*} \sum_{n\leq N}\tau(n)e\left(f(n)\right) ~\text{and}~\sum_{n\leq N}\mu^2(n)e\left(f(n)\right), \end{align*} for some polynomial $f$, where $\tau$ is the divisor function and $\mu$ is the M\"{o}bius function. We get non-trivial estimates when the leading co-efficient $\alpha$ of $f$ belongs to the minor arc.