Related papers: Counting non-standard binary representations

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…

Let $A = \{a_{1},a_{2},\dots{}\}$ $(a_{1} < a_{2} < \dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\in A)$. P. Erd\H{o}s, A. S\'ark\"ozy and V.…

Let $p/q$ ($p, q \in \mathbb{N}^*$) be a positive rational number such that $p > q^2$. We show that for any $\epsilon > 0$, there exists a set $A(\epsilon) \subset [0, 1[$, with finite border and with Lebesgue measure $< \epsilon$, for…

Let $f\colon\mathbb{N}\rightarrow\mathbb{C}$ be an arithmetic function and consider the Beatty set $\mathcal{B}(\alpha) = \lbrace\, \lfloor n\alpha \rfloor : n\in\mathbb{N} \,\rbrace$ associated to a real number $\alpha$, where…

We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{\nu,p}(a)=\sum_{n\geq 1} (an^p/2)^{-\nu} K_\nu(an^p)\qquad (a>0,\ 0\leq\nu<1)\] as the parameter $a\to 0+$, where $p$…

We have for positive integers $n$, $k$ and finite field $\mathbb{F}_q$, $c(n,k,q)$, as the number of simultaneous similarity classes of $k$-tuples of commuting $n\times n$ matrices over the $\mathbb{F}_q$. In this paper, it has been shown…

In this paper we study representations of real numbers in a numeral system with the base $a>1$ and alphabet (digits set) $A\equiv\{0,1,...,r\}$, $a-1<r\in N$ given by \[x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n}\equiv…

Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the "moments" F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a…

Let $\Z_m$ be the group of residue classes modulo $m$. Let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups of the group $\Z_m \times \Z_n$ and the number of its cyclic subgroups, respectively, where $m$ and $n$ are arbitrary…

Let $a(n)$ denote the number of non-isomorphic abelian groups with $n$ elements. In 1991 Ivi\'{c} proved an asymptotic formula of the sum $\sum_{n\leq x} a(n+ a(n)).$ In this paper, we will prove a sharper asymptotic formula for this sum.

Let $Q$ be a nondegenerate quadratic form on a vector space $V$ of even dimension $n$ over a number field $F$. Via the circle method or automorphic methods one can give good estimates for smoothed sums over the number of zeros of the…

For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ as a sum of $r = 2^n + 1$ summands, each of which is an $n$-th power of natural numbers $x_i$, $i = \overline{1,…

Let $f_1,\ldots,f_k : \mathbb{N} \rightarrow \mathbb{C}$ be multiplicative functions taking values in the closed unit disc. Using an analytic approach in the spirit of Hal\'{a}sz' mean value theorem, we compute multidimensional averages of…

We give asymptotic expansions for the moments of the $M_2$-rank generating function and for the $M_2$-rank generating function at roots of unity. For this we apply the Hardy-Ramanujan circle method extended to mock modular forms. Our…

Given an integer $g$ and also some given integers $m$ (sufficiently large) and $c_1,\dots, c_m$, we show that the number of all non-negative integers $n\le M$ with the property that there exist non-negative integers $k_1,\dots, k_m$ such…

A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right)…

Let $A_1$ and $A_2$ be randomly chosen subsets of the first $n$ integers of cardinalities $s_2\geq s_1 = \Omega(s_2)$, such that their sumset $A_1+A_2$ has size $m$. We show that asymptotically almost surely $A_1$ and $A_2$ are almost fully…

Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{n_1\cdots n_k\le x} F(n_1,\ldots,n_k)$, taken over the hyperbolic region $\{(n_1,\ldots,n_k)\in {\Bbb N}^k: n_1\cdots n_k\le x\}$, where $F:{\Bbb N}^k\to {\Bbb C}$ is a given…

In $1963$ Graham proved that every positive integer $n \ge 78$ can be written as a sum of distinct positive integers $a_1, a_2, \ldots, a_r$ for which $\frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_r}$ is equal to $1$. In the same…

We express some general type of infinite series such as $$ \sum^\infty_{n=1}\frac{F(H_n^{(m)}(z),H_n^{(2m)}(z),\ldots,H_n^{(\ell m)}(z))} {(n+z)^{s_1}(n+1+z)^{s_2}\cdots (n+k-1+z)^{s_k}}, $$ where $F(x_1,\ldots,x_\ell)\in\mathbb…