partition
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partition

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Distribution of powers of the partition function moduloj‘Jim L. BrownDepartment of Mathematical Sciences, Clemson University, Clemson, SC 296341Yingkun LiDepartment of Mathematics, California Institute of Technology, Pasadena, CA 91125AbstractIn this paper we study Newman’s conjecture for powers of the partitionfunction. While this conjecture is known for powers of primes ‘ that arenot exceptional for the power under consideration, it is an open problemfor primes. We settle this conjecture in many cases for smallpowers of the partition function by generalizing results of Ono and Ahlgren.It should be noted our method requires a case by case examination of eachpower and does not yield a general method for dealing with difierent powerssimultaneously.Key words: Newman’s Conjecture, partitions, modular forms2000 MSC: 11F33, 11P831. Introduction and Statement of ResultsA partition ofa positive integer n is a non-increasing sequence of positiveintegers whose sum is n. The partition function p(n) is deflned to be thenumber of partitions of n. By convention, p(0)=1 and p(n)=0 for n< 0.EulershowedthatthepartitionfunctionsatisflesthefollowinggeneratingEmail addresses: jimlb@clemson.edu (Jim L. Brown), yingkun@caltech.edu(Yingkun Li )1The second author was supported by a Summer Undergraduate Research Fellowshipat California Institute of Technology and the John and Maria La–n Trust.Preprint submitted to Elsevier December 2, 20086function relationship:1 1X Y ...

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Distribution of powers of the partition function modulo ` j Jim L. Brown Department of Mathematical Sciences, Clemson University, Clemson, SC 29634 Yingkun Li 1 Department of Mathematics, California Institute of Technology, Pasadena, CA 91125
Abstract In this paper we study Newman’s conjecture for powers of the partition function. While this conjecture is known for powers of primes ` that are not exceptional for the power under consideration, it is an open problem for exceptional primes. We settle this conjecture in many cases for small powers of the partition function by generalizing results of Ono and Ahlgren. It should be noted our method requires a case by case examination of each power and does not yield a general method for dealing with different powers simultaneously. Key words: Newman’s Conjecture, partitions, modular forms 2000 MSC: 11F33, 11P83
1. Introduction and Statement of Results A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n . The partition function p ( n ) is defined to be the number of partitions of n . By convention, p (0) = 1 and p ( n ) = 0 for n < 0. Euler showed that the partition function satisfies the following generating
Email addresses: jimlb@clemson.edu (Jim L. Brown), yingkun@caltech.edu (Yingkun Li ) 1 The second author was supported by a Summer Undergraduate Research Fellowship at California Institute of Technology and the John and Maria Laffin Trust.
Preprint submitted to Elsevier
December 2, 2008