Hardy–Littlewood Circle Method

Talukdar, Prabal (2025) Hardy–Littlewood Circle Method. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Prabal Talukdar (23MP021)) 23MP021_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (645kB)

Abstract

This thesis chapter investigates Waring’s problem using the Hardy–Littlewood circle method, a powerful analytic framework in additive number theory. We address the classical conjecture that every sufficiently large integer can be expressed as a sum of g(k) k-th powers, focusing on establishing an asymptotic formula for the number of such representations. The analysis divides the unit interval into major arcs (neighborhoods of rational numbers with small denominators) and minor arcs (the remaining regions). On major arcs, exponential sums are approximated using p-adic techniques and congruence analysis, leading to the singular series an arithmetic factor encoding local solution densities across prime power moduli. Simultaneously, the singular integral, derived from Gamma function asymptotics, quantifies the archimedean contribution. Weyl’s inequality and Hua’s lemma provide critical bounds for minor arc integrals, demonstrating their negligible contribution compared to major arcs. Key achievements include proving the convergence and positivity of the singular series, deriving sharp bounds for exponential sums, and synthesizing arithmetic and analytic components into a unified asymptotic formula. This work highlights the deep interplay between combinatorial number theory and complex analysis, while advancing classical results on representing integers as sums of powers. The methods developed here exemplify the circle method’s versatility in solving additive problems with both local and global constraints.

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