### 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 ≥N^{1/2+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

Original language | English |
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Pages (from-to) | 292-317 |

Number of pages | 26 |

Journal | Journal of Number Theory |

Volume | 73 |

Issue number | 2 |

DOIs | |

Publication status | Published - Dec 1998 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*73*(2), 292-317. https://doi.org/10.1006/jnth.1998.2288

**On the Parity of Additive Representation Functions.** / Nicolas, J. L.; Ruzsa, I.; Sárközy, A.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 73, no. 2, pp. 292-317. https://doi.org/10.1006/jnth.1998.2288

}

TY - JOUR

T1 - On the Parity of Additive Representation Functions

AU - Nicolas, J. L.

AU - Ruzsa, I.

AU - Sárközy, A.

PY - 1998/12

Y1 - 1998/12

N2 - 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

AB - 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

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UR - http://www.scopus.com/inward/citedby.url?scp=0000865092&partnerID=8YFLogxK

U2 - 10.1006/jnth.1998.2288

DO - 10.1006/jnth.1998.2288

M3 - Article

AN - SCOPUS:0000865092

VL - 73

SP - 292

EP - 317

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -