On the Parity of Additive Representation Functions

J. L. Nicolas, I. Z. Ruzsa, A. Sárközy

Research output: Contribution to journalArticle

47 Citations (Scopus)

Abstract

Let A be a set of positive integers,p(A,n) be the number of partitions ofnwith parts in A, andp(n)=p(N,n). It is proved that the number ofn≤Nfor whichp(n) is even is ≫N, while the number ofn≤Nfor whichp(n) is odd is ≥N1/2+o(1). Moreover, by using the theory of modular forms, it is proved (by J.-P. Serre) that, for allaandmthe number ofn, such thatn≡a(modm), andn≤Nfor whichp(n) is even is ≥cNfor any constantc, andNlarge enough. Further a set A is constructed with the properties thatp(A,n) is even for alln≥4 and its counting functionA(x) (the number of elements of A not exceedingx) satisfiesA(x)≫x/logx. Finally, we study the counting function of sets A such that the number of solutions ofa+a′=n,a,a′∈A,a<a′ is never 1 for largen.

Original languageEnglish
Pages (from-to)292-317
Number of pages26
JournalJournal of Number Theory
Volume73
Issue number2
DOIs
Publication statusPublished - Dec 1998

ASJC Scopus subject areas

  • Algebra and Number Theory

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