Weighted exponential sums and its applications

2024/08/04 by Nilanjan Bag, Bag, Nilanjan, Dwaipayan Mazumder +1
Mathematics · #11L03 #11L20 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2408.02020

openalex publication_date 2024/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∑_{n≤ N}τ(n)e(f(n)) ~and~∑_{n≤ N}μ²(n)e(f(n)), 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.

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