On the parity of additive representation functions

被引:48
作者
Nicolas, JL
Ruzsa, IZ
Sarkozy, A
机构
[1] Univ Lyon 1, Inst Girard Desargues, UPRES A 5028, F-69622 Villeurbanne, France
[2] Hungarian Acad Sci, Inst Math, H-1364 Budapest, Hungary
[3] Eotvos Lorand Univ, Dept Algebra & Number Theory, H-1088 Budapest, Hungary
基金
匈牙利科学研究基金会;
关键词
D O I
10.1006/jnth.1998.2288
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let A be a set of positive integers, p(A, n) be the number of partitions of n with parts in A, and p(n) = p(N, n). It is proved that the number of n less than or equal to N for which p(n) is even is much greater than root N while the number of n less than or equal to N for which p(n) is odd is greater than or equal to N1/2+o(1) Moreover, by using the theory of modular forms, it is proved (by J.-P. Serre) that, for all a and m the number of n, such that n equivalent to a (mod m), and n less than or equal to N for which p(n) is even is greater than or equal to c root N for any constant c, and N large enough. Further a set A is constructed with the properties that p(A, n) is even for all n greater than or equal to 4 and its counting function A(x) (the number of elements of A not exceeding x) satisfies A(x) much greater than x/logx. Finally, we study the counting: Function of sets A such that the number of solutions of a + a' = n, a, a' epsilon A, a < a' is never 1 for large n. (C) 1998 academic Press.
引用
收藏
页码:292 / 317
页数:26
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