Weighted exponential sums and its applications

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.