On the Boundedness of partial sums of multiplicative functions

• Autor(a) principal: Matheus Resende Guedes
• Data de Publicação: 2023
• Tipo de documento: Dissertação
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/59125
• topic: Erdős-Coons-Tao; Partial Sums; Multiplicative Functions; Boundedness; Analytic Number Theory; Matemática – Teses; Problema da discrepância de Erdős distintas; Teoria dos números - Teses.

In number theory the discrepancy of a function f: N→ C is defined as: \sup_{n, d} |\sum_{j=1}^n f(jd)|. The Erdos Discrepancy Problem asks whether the discrepancy of a function f: N → {-1, 1} is infinite. Tao showed in that this is indeed the case. Consequently, every totally multiplicative function that takes values in {-1, 1} has unbounded partial sums. This leads to a natural question: What happens when we consider multiplicative functions instead? Klurman provided a complete classification of multiplicative functions with bounded partial sums, a statement known as the Erdos–Coons–Tao conjecture. Another important related question is the study of what happens when we allow the codomain to be \C instead of {-1, 1}. In this case, there is no complete classification, but some results in this direction were studied by Aymone. The main goal of this dissertation is to understand the key steps in the proof of the Erdos-Coons-Tao conjecture and also investigate some related questions in Aymone's work when the codomain is C. The text is meant to be self-contained so we build all the necessary tools to understand the main results from the ground up, making this text accessible to anyone with basic undergraduate mathematics knowledge.