Divisor problems over special sequences

• Grant number: 22/16222-1
• Support Opportunities: Scholarships abroad - Research
• Effective date (Start): January 08, 2024
• Effective date (End): November 26, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: André Pierro de Camargo
• Grantee: André Pierro de Camargo
• Host Investigator: Igor Shparlinski
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil
• Research place: University of New South Wales (UNSW), Australia

Abstract

One of the oldest unsolved problems in Analytic Number Theory (the classical Dirichlet divisor problem) is determining the smallest order of the error term in the asymptotic expansion of $D(x) \ := \ \sum\limits_{n \leq x} \sum\limits_{d \mid n} 1$. Variants of this problem were considered throughout the years by imposing some conditions over the summation index $n$ or/and considering only the divisors $d$ of $n$ that fulfills some requirements. This project has three interdependent goals with different levels of complexity. The first and second goals are obtaining asymptotic expansions for $D$ when one takes $n \leq x$ in some subsets $K$ of the set $\mathcal A_k$ of $k$-free integers. We are interested mainly in the cases $K = \mathcal A_k$ (first goal) and $K = \mathcal A_k \cap \{ j \in \ \mathbb N : j \equiv a \ (mod) \ q \}$ (second goal). After this is accomplished, we will be concerned in obtaining analogous results when $\mathcal A_k$ is replaced by some classes of $B$-free integers. The third goal of this project is using the tools and skills developed in the previous tasks to understand an open conjecture connecting a complex divisor problem with the statistical behavior of the fractional parts of the sequence $(\alpha n^2)_{n \geq 1}, \ \alpha \ \notin \mathbb Q$. (AU)