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We consider a wide class of summatory functions F{f;N,p^m}=\sum_{k\leq N}f(p^m k), m\in \mathbb Z_+\cup {0}, associated with the multiplicative arithmetic functions f of a scaled variable k\in \mathbb Z_+, where p is a prime number.…

Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these results to the number of parts divisible by k in all the…

We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a…

Given an integer $n \ge 2$, its prime factorization is expressed as $n= \prod_{i=1}^s p_i^{a_i}$. We define the function $f(n)$ as the smallest positive integer such that $f(n)!$ is divisible by $n$. The main objective of this paper is to…

We study the series s(n,x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson's theorem we show that the integer part of s(n,x) for x = Pi/2 is the number of primes…

This paper presents a reformulation of the Leibniz product rule as a finite sum that expresses the fractional derivative of the product of two differentiable functions. This paper then proves the cases for when the product consists of an…

We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these…

Let $D$ be an open disk of radius $\le 1$ in $\mathbb C$, and let $(\epsilon_n)$ be a sequence of $\pm 1$. We prove that for every analytic function $f: D \to \mathbb C$ without zeros in $D$, there exists a unique sequence $(\alpha_n)$ of…

A numerical semigroup $S$ is an additively-closed set of non-negative integers, and a factorization of an element $n$ of $S$ is an expression of $n$ as a sum of generators of $S$. It is known that for a given numerical semigroup $S$, the…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a prime power and $n$ be a positive integer. In this paper, we explore the factorization of $f(x^{n})$ over $\mathbb{F}_q$, where $f(x)$ is an irreducible polynomial…

In this paper we review recently established results on the asymptotic behaviour of the trigonometric product $P_n(\alpha) = \prod_{r=1}^n |2\sin \pi r \alpha|$ as $n\to \infty$. We focus on irrationals $\alpha$ whose continued fraction…

Let G be a piecewise constant $n\times n$ matrix function which is defined on a smooth closed curve $\Gamma$ in the complex sphere and which has m jumps. We consider the problem of determining the partial indices of the factorization of the…

Let $n=\prod_p p^{\nu_p(n)}$ denote the canonical factorization of $n\in \N$. The binomial convolution of arithmetical functions $f$ and $g$ is defined as $(f\circ g)(n)=\sum_{d\mid n} (\prod_p \binom{\nu_p(n)}{\nu_p(d)}) f(d)g(n/d),$ where…

We extend a factorization due to Krein to arbitrary analytic functions from the upper half-plane to itself. The factorization represents every such function as a product of fractional linear factors times a function which, generally, has…

We study the (variable-length) codes of the form X u {a^n}, where X c a*wa* and |X| = n. We extend various notions and results from factorizations of cyclic groups theory to this type of codes. In particular, when n is the product of at…

For numerical semigroups with three generators, we study the asymptotic behavior of weighted factorization lengths, that is, linear functionals of the coefficients in the factorizations of semigroup elements. This work generalizes many…

We give an overview of combinatoric properties of the number of ordered $k$-factorizations $f_k(n,l)$ of an integer, where every factor is greater or equal to $l$. We show that for a large number $k$ of factors, the value of the cumulative…

We consider GL_n(F_q)-analogues of certain factorization problems in the symmetric group S_n: rather than counting factorizations of the long cycle (1, 2, ..., n) given the number of cycles of each factor, we count factorizations of a…

Let $\mathbb{P}$ denote the set of primes and $\mathcal{N}\subset \mathbb{N}$ be a set with arbitrary weights attached to its elements. Set $\mathfrak{p}_{\mathcal{N}}(n)$ to be the restricted partition function which counts partitions of…

Graphical models represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function, is the main inference challenge relevant to multiple statistical and…