Related papers: Sieve functions in arithmetic bands, II

For two arithmetical functions $f$ and $g$, we study the convolution sum of the form $\sum_{n \le N} f(n) g(n+h)$ in the context of its asymptotic formula with explicit error terms. Here we introduce the concept of finite Ramanujan…

Avikainen showed that, for any $p,q \in [1,\infty)$, and any function $f$ of bounded variation in $\mathbb{R}$, it holds that $\mathbb{E}[|f(X)-f(\widehat{X})|^{q}] \leq C(p,q) \mathbb{E}[|X-\widehat{X}|^{p}]^{\frac{1}{p+1}}$, where $X$ is…

Let $R$ be a finite commutative ring with $1\ne 0$. The set $\mathcal{F}(R)$ of polynomial functions on $R$ is a finite commutative ring with pointwise operations. Its group of units $\mathcal{F}(R)^\times$ is just the set of all…

Sign changes in sums of arithmetic functions and their inverses are a subtle topic with room to grow new results. Suppose that $S_f(x) := \sum_{n \leq x} f(n)$ is the summatory function of some arithmetic function $f$ such that $f(1) \neq…

In this manuscript, we investigate some properties of certain counting functions, associated to the ergodic sums computed along the periodic orbits of the skew-product map, related to a finitely generated rational semigroup. To be precise,…

In this paper we establish a new formula for the arithmetic functions that verify $ f(n) = \sum_{d|n} g(d)$ where $g$ is also an arithmetic function. We prove the following identity, $$\forall n \in \mathbb{N}^*, \ \ \ f(n) = \sum_{k=1}^n…

Let $\mathcal{P}$ be the set of the primes. We consider a class of random multiplicative functions $f$ supported on the squarefree integers, such that $\{f(p)\}_{p\in\mathcal{P}}$ form a sequence of $\pm1$ valued independent random…

For a generalized permutohedron $Q$ the enumerator $F(Q)$ of positive lattice points in interiors of maximal cones of the normal fan $\Sigma_Q$ is a quasisymmetric function. We describe this function for the class of nestohedra as a Hopf…

Let F be a flat vector bundle over a compact Riemannian manifold M and let f be a Morse function. Let g be a smooth Euclidean metric on F, let g_t=e^{-2tf}g and let \rho(t) be the Ray-Singer analytic torsion of F associated to the metric…

We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for…

We explore separable property of the joint numerical range $G(\Bbb R^n)$ of a special class of quadratic functions and apply it to solving the smallest enclosing ball (SEB) problem which asks to find a ball $B(a,r)$ in $\Bbb R^n$ with…

The minimal and the maximal sections $\wedge_f,\vee\!_f:X\to\overline{\mathbb R}$ of a function $f:X\times Y\to\overline{\mathbb R}$ are defined by $\wedge_f(x)=\inf\limits_{y\in Y}f(x,y)$ and $\vee\!\!_f(x)=\sup\limits_{y\in Y}f(x,y)$ for…

In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalise strong $q$-additivity and -multiplicativity, respectively. We show that…

In the field of radial basis functions mathematicians have been endeavouring to find infinitely differentiable and compactly supported radial functions. This kind of functions are extremely important for some reasons. First, its…

Let $[t]$ be the integral part of the real number $t$.The aim of this short note is to study the distribution of elements of the set $\mathcal{S}(x) := \{[\frac{x}{n}] : 1\le n\le x\}$ in the arithmetical progression $\{a+dq\}_{d\ge 0}$.Our…

A convex function $f:[a,b]\to\mathbb{R}$ satisfies the so-called Hermite-Hadamard inequality $$ f\left(\frac{a+b}{2}\right)\leq \frac{1}{b-a}\int_a^{b}f(t)dt\leq \frac{f(a)+f(b)}{2}. $$ Motivated by the above estimates, in this paper we…

In this paper we describe all differentiable functions $\varphi,\psi\colon E\to\mathbb{R}$ satisfying the functional-differential equation \begin{equation*} [\varphi(y) - \varphi(x)]\psi '\bigl(h(x,y)\bigr) = [\psi(y) - \psi(x)]\varphi…

A function on the real line is called regulated if it has a left limit and a right limit at each point. If $f$ is a Schwartz distribution on the real line such that $f=F'$ (distributional or weak derivative) for a regulated function $F$…

We here revisit Fourier analysis on the Heisenberg group H^d. Whereas, according to the standard definition, the Fourier transform of an integrable function f on H^d is a one parameter family of bounded operators on L 2 (R^d), we define (by…

In previous work by Beer and Levi [8, 9], the authors studied the oscillation $\Omega (f,A)$ of a function $f$ between metric spaces $\langle X,d \rangle$ and $\langle Y,\rho \rangle$ at a nonempty subset $A$ of $X$, defined so that when $A…