A New Lower Bound on the Number of Odd Values of the Ordinary Partition Function

Eichhorn, Dennis

doi:10.1007/s00026-009-0030-0

Cited by 7 publications

(5 citation statements)

References 13 publications

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“…for n ≤ x, due to Bellaïche, Green, and Soundararajan [4] (who have improved results of several other authors; see, just as a sample, [1,5,12,23,26,34]), is given by √ x log log x , for x → ∞. (Tangentially, we know from [5] that the number of n ≤ x such that p(n) is even is at least of order √ x log log x; notice also that, unlike for the odd values, an asymptotic lower bound of order √ x for the even values is very easy to obtain.…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

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On the Density of the Odd Values of the Partition Function

2018

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The purpose of this note is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function p(n) is equidistributed modulo 2.Our main result will relate the densities, say δ t , of the odd values of the t-multipartition functions p t (n), for several integers t. In particular, we will show that if δ t > 0 for some 7,11,13,17,19,23, 25}, then (assuming it exists) δ 1 > 0; that is, p(n) itself is odd with positive density. Notice that, currently, the best unconditional result does not even imply that p(n) is odd for √ x values of n ≤ x. In general, we conjecture that δ t = 1/2 for all t odd, i.e., that similarly to the case of p(n), all multipartition functions are in fact equidistributed modulo 2.Our arguments will employ a number of algebraic and analytic methods, ranging from an investigation modulo 2 of some classical Ramanujan identities and several other eta product results, to a unified approach that studies the parity of the Fourier coefficients of a broad class of modular form identities recently introduced by Radu.

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Section: Introduction and Main Resultsmentioning

confidence: 96%

“…for n ≤ x, due to Bellaïche, Green, and Soundararajan [4] (who have improved results of several other authors; see, just as a sample, [1,5,12,23,26,34]), is given by…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

On the Density of the Odd Values of the Partition Function

2018

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“…However, despite important efforts by many researchers using a variety of combinatorial, algebraic or analytic tools, this conjecture still appears to be out of reach for today's mathematics. The current best lower bounds for f 0 p (x) and f 1 p (x) are due to J. Bellaïche and J.-L. Nicolas [5], who refined results by S. Ahlgren, D. Eichhorn, K. Ono, and J.-P. Serre, among many others (see, as a sample, [1,10,14,16,21]). Namely, using the theory of modular forms, Bellaïche and Nicolas proved:…”

Section: Introduction and Past Workmentioning

confidence: 87%

On the number of odd values of the Klein j-function and the cubic partition function

Zanello

2015

Journal of Number Theory

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In this note, using entirely algebraic or elementary methods, we determine a new asymptotic lower bound for the number of odd values of one of the most important modular functions in number theory, the Klein j-function. Namely, we show that the number of integers n ≤ x such that the Klein j-function -or equivalently, the cubic partition function -is odd is at least of the order of √ x log log x log x , for x large. This improves recent results of Berndt-Yee-Zaharescu and Chen-Lin, and approaches significantly the best lower bound currently known for the ordinary partition function, obtained using the theory of modular forms. Unlike many works in this area, our techniques to show the above result, that have in part been inspired by some recent ideas of P. Monsky on quadratic representations, do not involve the use of modular forms. Then, in the second part of the article, we show how to employ modular forms in order to slightly refine our bound. In fact, our brief argument, which combines a recent result of J.-L. Nicolas and J.-P. Serre with a classical theorem of J.-P. Serre on the asymptotics of the Fourier coefficients of certain level 1 modular forms, will more generally apply to provide a lower bound for the number of odd values of any positive power of the generating function of the partition function.

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“…Currently, the best result, which has improved on the work of multiple authors (see for example [1,11,14,25,29]), is due to Bellaïche, Green, and Soundararajan [10] and states that the number of odd values of p(n) for n ≤ x has at least the order of √ x log log x , for x → ∞. In fact, their bound holds for any t-multipartition function p t (n), thus improving on a result of the second author (cf.…”

Section: Introductionmentioning

confidence: 99%

On the density of the odd values of the partition function, II: An infinite conjectural framework

Judge¹,

Zanello²

2017

Preprint

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We continue our study of a basic but seemingly intractable problem in integer partition theory, namely the conjecture that p(n) is odd exactly 50% of the time. Here, we greatly extend on our previous paper by providing a doubly-indexed, infinite framework of conjectural identities modulo 2, and show how to, in principle, prove each such identity. However, our conjecture remains open in full generality.A striking consequence is that, under suitable existence conditions, if any t-multipartition function is odd with positive density and t ≡ 0 ( mod 3), then p(n) is also odd with positive density. These are all facts that appear virtually impossible to show unconditionally today.Our arguments employ a combination of algebraic and analytic methods, including certain technical tools recently developed by Radu in his study of the parity of the Fourier coefficients of modular forms.Hence, p t (n) denotes the number of t-multipartitions (also called t-colored partitions) of n.The goal of this paper is to provide additional insight into the long-standing question of estimating the density of the odd values of the partition function. As Paul Monsky once pointedly stated [22], "the best minds of our generation haven't gotten anywhere with understanding the parity of p(n)." It is a famous conjecture of Parkin and Shanks [30]

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A New Lower Bound on the Number of Odd Values of the Ordinary Partition Function

Cited by 7 publications

References 13 publications

On the Density of the Odd Values of the Partition Function

On the Density of the Odd Values of the Partition Function

On the number of odd values of the Klein j-function and the cubic partition function

On the density of the odd values of the partition function, II: An infinite conjectural framework

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