Related papers: The divisor function over integers with a missing …

We consider the sum of digits functions for both base phi, and for the Zeckendorf expansion of the natural numbers. For both sum of digits functions we present morphisms on infinite alphabets such that these functions viewed as infinite…

We compute the variances of sums in arithmetic progressions of generalised k-divisor functions related to certain L-functions in $\mathbb{F}_q(t)$, in the limit as $q\to\infty$. This is achieved by making use of recently established…

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their…

Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. In this paper, we prove that $$\sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+\mathcal{O}_{\varepsilon,c}…

How to handle division in systems that compute with logical formulas involving what would otherwise be polynomial constraints over the real numbers is a surprisingly difficult question. This paper argues that existing approaches from both…

Arithmetic constraints on integer intervals are supported in many constraint programming systems. We study here a number of approaches to implement constraint propagation for these constraints. To describe them we introduce integer interval…

For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $\Phi\left(n\right)$ denote, respectively, the number of subsets $A$ of…

We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.

For any finite set of integers X, define its sumset X+X to be {x+y: x, y in X}. In a recent paper, Martin and O'Bryant investigated the distribution of |A+A| given the uniform distribution on subsets A of {0, 1, ..., n-1}. They also…

We show that minimizing a convex function over the integer points of a bounded convex set is polynomial in fixed dimension.

The summation arithmetic functions with asymptotically independent summands are studied in the paper. We prove statements about the condition under which the summation arithmetic functions have asymptotically independent summands. It is…

Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals…

The theorem below gives another way of computing the distribution prime counting function without using recursion and the values of Prime numbers

For $r \in \mathbb{R}, r> 1$ and $n \in \mathbb{Z}^+$, the divisor function $\sigma_{-r}$ is defined by $\sigma_{-r}(n) := \sum_{d \vert n} d^{-r}$. In this paper we show the number $C_r$ of connected components of…

In this paper we introduce a technique to determine the sumset $A+A$, where $A$ is the indicator function of the 0's occurring in a fixed point $x$ of a substitution on the alphabet $\{0,1\}$.

Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define…

Let $d(n)$ denote the Dirichlet divisor function. Define \begin{equation*} \mathcal{S}_{k}(x)=\sum_{\substack{1\leqslant n_1,n_2,n_3 \leqslant x^{1/2} \\ 1\leqslant n_4\leqslant x^{1/k} }} d(n_1^2+n_2^2+n_3^2+n_4^k), \qquad 3\leqslant k\in…

In this paper we give a method, based on the characteristic function of a set, to solve some difficult problems of set theory in undergraduate research.

Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial…

We study sums with multiplicative functions that take values over a non-homogenous Beatty sequence. We then apply our result in a few special cases to obtain asymptotic formulas such as the number of integers in a Beatty sequence…