Slim exceptional sets for sums of four squares

TD Wooley

Research output: Contribution to journalArticle (Academic Journal)

Abstract

Given that available technology permits one to establish that almost all natural numbers satisfying appropriate congruence conditions are represented as the sum of three squares of prime numbers, one expects strong estimates to be attainable for exceptional sets in the analogous problem involving sums of four squares of primes. Let E(N) denote the number of positive integers not exceeding N that are congruent to 4 modulo 24, yet cannot be written as the sum of four squares of prime numbers. A method is described that shows that for each positive number , one has , thereby exploiting effectively the ‘excess’ fourth square of a prime so as to improve the recent bound due to J. Liu and M.-C. Liu. It transpires that the ideas underlying this progress permit estimates for exceptional sets in a variety of additive problems to be significantly slimmed whenever sufficiently many excess variables are available. Such ideas are illustrated for several additional problems involving sums of four squares.
Translated title of the contributionSlim exceptional sets for sums of four squares
Original languageEnglish
Pages (from-to)1 - 21
Number of pages21
JournalProceedings of the London Mathematical Society
Volume85 (1)
DOIs
Publication statusPublished - Jul 2002

Bibliographical note

Publisher: London Math Soc

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