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 dissertation 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 $2$. We provide a doubly-indexed, infinite family of conjectural identities in the ring of series $\Z_2[[q]]$, which relate $p(n)$ with suitable $t$-multipartition functions, and show how to, in principle, prove each such identity. We will exhibit explicit proofs for $32$ of our identities. However, the conjecture remains open in full generality. A striking consequence of these conjectural identities is that, under suitable existence conditions, for any $t$ coprime to $3$, if the $t$-multipartition function is odd with positive density, then $p(n)$ is also odd with positive density. Additionally if \emph{any} $t$-multipartition function is odd with positive density, then either $p(n)$ or the $3$-multipartition function (or both) are odd with positive density. All of these facts 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

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

Search for New Physics in Rare Top Decays: ttˉt \bar t Spin Correlations and Other Observables

In this paper we study new-physics contributions to the top-quark decay $t \to b \bar b c$. We search for ways of detecting such new physics via measurements at the LHC. As top quarks are mainly produced at the LHC in $t \bar t$ production via gluon fusion, we analyze the process $gg \to t \bar t \to (b \bar b c) (\bar b \ell \bar \nu)$. We find six observables that can be used to reveal the presence of new physics in $t \to b \bar b c$. Three are invariant mass-squared distributions involving two of the final-state particles in the top decay, and three are angular correlations between the final-state quarks coming from the $t$ decay and the $\ell^-$ coming from the $\bar t$ decay. The angular correlations are related to the $t \bar t$ spin correlation.Comment: Published versio

Search for new physics in rare top decays: t¯t spin correlations and other observables

In this paper we study new-physics contributions to the top-quark decay t → bbc. We search for ways of detecting such new physics via measurements at the LHC. As top quarks are mainly produced at the LHC in tt production via gluon fusion, we analyze the process gg → tt → bbc bℓν . We find six observables that can be used to reveal the presence of new physics in t → bbc. Three are invariant mass-squared distributions involving two of the final-state particles in the top decay, and three are angular correlations between the final-state quarks coming from the t decay and the ℓ − coming from the t decay. The angular correlations are related to the t t spin correlation.Instituto de Física La Plat

Search for new physics in rare top decays: t¯t spin correlations and other observables

In this paper we study new-physics contributions to the top-quark decay t → bbc. We search for ways of detecting such new physics via measurements at the LHC. As top quarks are mainly produced at the LHC in tt production via gluon fusion, we analyze the process gg → tt → bbc bℓν . We find six observables that can be used to reveal the presence of new physics in t → bbc. Three are invariant mass-squared distributions involving two of the final-state particles in the top decay, and three are angular correlations between the final-state quarks coming from the t decay and the ℓ − coming from the t decay. The angular correlations are related to the t t spin correlation.Instituto de Física La Plat