Weighted exponential sums and its applications
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911977070133248 |
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| author | Bag, Nilanjan Mazumder, Dwaipayan |
| author_facet | Bag, Nilanjan Mazumder, Dwaipayan |
| contents | 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}τ(n)e\left(f(n)\right) ~\text{and}~\sum_{n\leq N}μ^2(n)e\left(f(n)\right), \end{align*} for some polynomial $f$, where $τ$ is the divisor function and $μ$ is the Möbius function. We get non-trivial estimates when the leading co-efficient $α$ of $f$ belongs to the minor arc. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02020 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted exponential sums and its applications Bag, Nilanjan Mazumder, Dwaipayan Number Theory 11L03, 11L20 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}τ(n)e\left(f(n)\right) ~\text{and}~\sum_{n\leq N}μ^2(n)e\left(f(n)\right), \end{align*} for some polynomial $f$, where $τ$ is the divisor function and $μ$ is the Möbius function. We get non-trivial estimates when the leading co-efficient $α$ of $f$ belongs to the minor arc. |
| title | Weighted exponential sums and its applications |
| topic | Number Theory 11L03, 11L20 |
| url | https://arxiv.org/abs/2408.02020 |