Related papers: Log-convexity and the overpartition 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…

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3- and 4-dissections of the generating function for $\overline{p}(n)$ and derived a number of…

Consider the eigenvalue problem of a linear second order elliptic operator: \begin{equation} \nonumber -D\Delta \varphi -2\alpha\nabla m(x)\cdot \nabla\varphi+V(x)\varphi=\lambda\varphi\ \ \hbox{ in }\Omega, \end{equation} complemented by…

In this paper, we prove that the discrete Copson inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 49-51) of one-dimension in general cases admits an improvement. In fact we study the…

Let $1 \leq p < q < \infty$. We show that \[ \sup{\frac{\left\| D\right\|_{\mathcal{H}_{q}}}{\left\| D\right\|_{\mathcal{H}_{p}}}} = \exp{\left( \frac{\log{x}}{\log{\log{x}}} \left(\log{\sqrt{\frac{q}{p}}} +…

Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals, which are called the Catalan-Larcombe-French sequence $\{P_n\}_{n\geq 0}$ and the Fennessey-Larcombe-French sequence $\{V_n\}_{n\geq…

Denote by $\Delta$ the Laplacian and by $\Delta_\infty $ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $\Delta v\Delta_\infty v$: for every $v\in C^\infty$, $$\ | { |D^2vDv|^2} - {\Delta v…

Many papers have studied inequalities for Andrews and Paule's broken $k$-diamond partition function $\Delta_{k}(n)$ when $k=1$ or $2$. In this paper, we derive an exact formula for $\Delta_{k}(n)$ when $k\geq 1$. Building on this result, we…

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\left\{xy = n\right\}$ has between $\Omega(n^{1/3})$ and $O(n^{1/3} \log n)$ vertices. The same bounds apply to any hyperbola with rational…

We prove that the number $q(n)$ of partitions into distinct parts is log-concave for $n \geq 33$ and satisfies the higher order Tur\'an inequalities for $n\geq 121$ conjectured by Craig and Pun. In doing so, we establish explicit error…

We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\Delta u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in…

Let $a>1$ be an integer. Denote by $l_a(p)$ the multiplicative order of $a$ modulo primes $p$. We prove that if $\frac{x}{\log x\log\log x}=o(y)$, then $$\frac 1 y \sum_{a\leq y}\sum_{p\leq x}\frac{1}{l_a(p)}=\log x + C\log\log…

We prove a lower bound of exp(-C (log(2/alpha))^7)N^{k-1} to the number of solutions of an invariant equation in k variables, contained in a set of density alpha. Moreover, we give a Behrend-type construction for the same problem with the…

Let $p(n)$ denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order $N$ (for any fixed positive integer $N$) along with estimates for error bounds for the shifted quotient of the partition…

In this paper we consider a weighted version of one dimensional discrete Hardy's Inequality on half-line with power weights of the form $n^\alpha$. Namely we consider: \begin{equation} \sum_{n=1}^\infty |u(n)-u(n-1)|^2 n^\alpha \geq…

Let $n\geq 3$, $\alpha$, $\beta\in\mathbb{R}$, and let $v$ be a solution $\Delta v+\alpha e^v+\beta x\cdot\nabla e^v=0$ in $\mathbb{R}^n$, which satisfies the conditions $\lim_{R\to\infty}\frac{1}{\log R}\int_{1}^{R}\rho^{1-n}…

We consider the real $\beta$-ensemble (or 1D log-gas) of dimension $N$ in the high-temperature regime, \textit{i.e.} where the inverse temperature $\beta$ scales as $N\beta=2P$ with $P$ a fixed positive parameter. We establish the large-$N$…

We obtain a quantitative estimate of Bergman distance when $\Omega \subset \mathbb{C}^n$ is a bounded domain with log-hyperconvexity index $\alpha_l(\Omega)>\frac{n-1+\sqrt{(n-1)(n+3)}}{2}$, as well as the $A^2(\log A)^q$-integrability of…

Let $A\subset\left\{ 1,\dots,N\right\} $ be a set of prime numbers containing no non-trivial arithmetic progressions. Suppose that $A$ has relative density $\alpha=|A|/\pi(N)$, where $\pi(N)$ denotes the number of primes in the set $\left\{…

Recently, Shen (2016) and Alanazi et al. (2016) studied the arithmetic properties of the $\ell$-regular overpartition function $\overline{A}_\ell (n)$, which counts the number of overpartitions of $n$ into parts not divisible by $\ell$. In…