Related papers: Partition-theoretic formulas for arithmetic densit…

The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper we consider the analogous multiplicative setting of the cyclic group $\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$, and…

We define the notion of couple density $(D, \mathbf b)$ where $D$ is a non-empty subset of $\mathbb Z^{m}$ and $ \mathbf b$ a fixed element in $\{0, \cdots, q-2\}^{m};$ We determine a minimum in terms of the density of the couple…

The partition algebras are algebras of diagrams (which contain the group algebra of the symmetric group and the Brauer algebra) such that the multiplication is given by a combinatorial rule and such that the structure constants of the…

We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators ($q$-numbers) of a…

We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…

Consider a class of probability distributions which is dense in the space of all probability distributions on $\mathbb{R}^{d}$ with respect to weak convergence, for every $d\in\mathbb{N}$. Then, we construct various explicit classes of…

Let $A$ be a subset of primes up to $x$. If we assume $A$ is well-distributed (in the Siegel-Walfisz sense) in any arithmetic progressions to moduli $q\leqslant(\log x)^c$ for any $c>0$, then the sumset $A+A$ has density 1/2 in the natural…

The partition function $p(n)$, which counts the number of partitions of a positive integer $n$, is widely studied. Here, we study partition functions $p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain congruence…

Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under…

Let N_g(d) be the set of primes p such that the order of g modulo p is divisible by a prescribed integer d. Wiertelak showed that this set has a natural density and gave a rather involved explicit expression for it. Let N_g(d)(x) be the…

We compute the one-level density of zeros of order $\ell$ Dirichlet $L$-functions over function fields $\mathbb{F}_q[t]$ for $\ell=3,4$ in the Kummer setting ($q\equiv1\pmod{\ell}$) and for $\ell=3,4,6$ in the non-Kummer setting…

We compute the dimension $d_{n,r}(q) = \dim(\IR_q^r)$ of the defining module $\IR_q^r$ for the $q$-partition algebra. This module comes from $r$-iterations of Harish-Chandra restriction and induction on $\GL_n(\FF_q)$. This dimension is a…

In this paper we refine a weighted partition identity of Alladi. We write explicit formulas of generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results…

Several measures for the density of sets of integers have been proposed, such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of…

We propose a new fractional statistics for arbitrary dimensions, based on an extension of Pauli's exclusion principle, to allow for finite multi-occupancies of a single quantum state. By explicitly constructing the many-body Hilbert space,…

We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. This…

This article addresses a fundamental problem faced by the ab initio community: the lack of an effective formalism for the rapid exploration and exchange of new methods. To rectify this, we introduce a novel, basis-set independent,…

Schur's partition theorem states that the number of partitions of n into distinct parts congruent 1, 2 (mod 3) equals the number of partitions of n into parts which differ by >= 3, where the inequality is strict if a part is a multiple of…

The set of integers which can be written as the sum of four prime cubes has lower density at least $0.009664$. This improves earlier bounds of $0.003125$ by Ren and $0.005776$ by Liu.

Building on the notion of $q$-integral introduced by Thomae in 1869, we introduce $q$-order statistics (that, is $q$-analogues of the classical order statistics, for $0<q<1$) which arise from dependent and not identically distributed…