On the Parity of Additive Representation Functions Original Research Article

Let image be a set of positive integers,p(image, n) be the number of partitions ofnwith parts in image, andp(n)=p(N, n). It is proved that the number ofnless-than-or-equals, slantNfor whichp(n) is even is much greater-thanimage, while the number ofnless-than-or-equals, slantNfor whichp(n) is odd is greater-or-equal, slantedN1/2+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(mod m), andnless-than-or-equals, slantNfor whichp(n) is even is greater-or-equal, slantedcimagefor any constantc, andNlarge enough. Further a set image is constructed with the properties thatp(image, n) is even for allngreater-or-equal, slanted4 and its counting functionA(x) (the number of elements of image not exceedingx) satisfiesA(x)much greater-thanx/log x. Finally, we study the counting function of sets image such that the number of solutions ofa+a′=n,a,a′set membership, variantimage,a