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

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 $t\not \equiv 0$ (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.Comment: 14 pages. To appear in the J. of Number Theor

On the density of the odd values of the partition function

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 $\delta_t$, of the odd values of the $t$-multipartition functions $p_t(n)$, for several integers $t$. In particular, we will show that if $\delta_t>0$ for some $t\in \{5,7,11,13,17,19,23,25\}$, then (assuming it exists) $\delta_1>0$; 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 $\sqrt{x}$ values of $n\le x$. In general, we conjecture that $\delta_t=1/2$ 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.Comment: Several changes with respect to the 2015 version. 18 pages. To appear in the Annals of Combinatoric

Non-existence of Ramanujan congruences in modular forms of level four

Ramanujan famously found congruences for the partition function like p(5n+4) = 0 modulo 5. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Gamma_{1}(4) which is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.Comment: 19 page

The Non-existence and Scarcity of Congruences for Partitions

We investigate Ramanujan congruences for the function $\overline{t}(n)$, which counts the overpartitions of $n$ with restricted odd differences, and the existence of certain congruences of for $p_r(n)$. In particular, we show that one Ramanujan congruence exists for $\overline{t}(n)$ and that congruences of the form $p_r(\ell Q^m + \beta)$ for $\ell, Q$ prime and $m = 1,2$ appear to be scarce. The method for both results uses the theory of modular forms. In the former case, a more general theorem which bounds the number of primes possible for Ramanujan congruences in certain eta-quotients is proved, which generalizes work done by Jonah Sinick. In the latter case, we develop several necessary conditions for the existence of such congruences, which generalizes the work of Ahlgren et. al. for $p(n)$

Effective Congruences for Mock Theta Functions

Let M(q)=∑c(n)qn role= presentation style= box-sizing: border-box; max-height: none; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial; position: relative; \u3eM(q)=∑c(n)qnM(q)=∑c(n)qn be one of Ramanujan’s mock theta functions. We establish the existence of infinitely many linear congruences of the form: c(An+B)≡0 (modlj) role= presentation style= box-sizing: border-box; max-height: none; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(34, 34, 34); font-family: Arial; position: relative; \u3ec(An+B)≡0 (modlj)c(An+B)≡0 (modlj) where A is a multiple of l and an auxiliary prime, p. Moreover, we give an effectively computable upper bound on the smallest such p for which these congruences hold. The effective nature of our results is based on the prior works of Lichtenstein [1] and Treneer [2]

Coefficients of modular forms and applications to partition theory

We begin with an overview of the theory of modular forms as well as some relevant sub-topics in order to discuss three results: the first result concerns positivity of self-conjugate t-core partitions under the assumption of the Generalized Riemann Hypothesis; the second result bounds certain types of congruences called Ramanujan congruences for an infinite class of eta-quotients - this has an immediate application to a certain restricted partition function whose congruences have been studied in the past; the third result strengthens a previous result that relates weakly holomorphic modular forms to newforms via p-adic limits