Related papers: On the representation functions of certain numerat…

Let $d_{\alpha, \beta}(n)=\sum\limits_{\substack{n=kl \alpha l<k\leq\beta l}}1$ be the number of ways of factoring n into two almost equal integers. For rational numbers $0<\alpha <\beta $, we consider the following Zeta function…

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…

For fixed natural numbers $r$ and $s$, where $2\leq s \leq r$, we consider a representation of numbers from the interval $[0;\frac{r}{s-1}]$ obtained by encoding numbers by means of the alphabet $A=\{0,1,...,r\}$ via the expansion…

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an…

Let $a, b,c $ and $k$ be positive integers such that $1\leq a\leq b,a<c<2(a+b), c\ne b$ and $(a,b,c)=1$. Define the arithmetic function $f_k(a,b;c;n)$ by $$ \sum_{n=1}^{\infty}\frac{f_k(a,b;c;n)}{n^s}=\frac{\zeta (as)\zeta…

Let $I(b,d,k)$ be the subseries of the harmonic series keeping the integers having exactly $k$ occurrences of the digit $d$ in base $b$. We prove the existence of an asymptotic expansion to all orders in descending powers of $b$, for fixed…

The approximation constant $\lambda_{k}(\zeta)$ is defined as the supremum of real $\eta$ such that $\Vert \zeta^{j}x\Vert\leq x^{-\eta}$ for $1\leq j\leq k$ has infinitely many integer solutions $x$. Here $\Vert.\Vert$ denotes the distance…

We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice $\Lambda \subseteq \mathbb{R}^d$ inside the $d$-sphere of radius $R$. In contrast to…

Let us denote by $Z_{b}(n)$ the number of trailing zeroes in the base b expansion of $n!$. In this paper we study the connection between the expression of $\vartheta(b):=\lim_{n\to \infty}Z_{b}(n)/n$ in base $b$, and that of $Z_{b}(b^{k})$.…

Let $(\alpha,\mathcal{N}_{\alpha})$ and $(\beta,\mathcal{N}_{\beta})$ be two canonical number systems for an imaginary quadratic number field $K$ such that $\alpha$ and $\beta$ are multiplicatively independent. We provide an effective lower…

A numeral system is an infinite sequence of different closed normal $\lambda$-terms intended to code the integers in $\lambda$-calculus. H. Barendregt has shown that if we can represent, for a numeral system, the functions : Successor,…

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 present several formulae for the large $t$ asymptotics of the Riemann zeta function $\zeta(s)$, $s=\sigma+i t$, $0\leq \sigma \leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical…

We refine a remark of Steinerberger (2024), proving that for $\alpha \in \mathbb{R}$, there exists integers $1 \leq b_{1}, \ldots, b_{k} \leq n$ such that \[ \left\| \sum_{j=1}^k \sqrt{b_j} - \alpha \right\| = O(n^{-\gamma_k}), \] where…

In the paper we describe some properties of function $$ y=r(x)=\lim_{n\to\infty}\frac{1}{n}\sum^{\infty}_{k=1}\alpha_k(x), \text{ where } x=\sum^{\infty}_{k=1}\alpha_k(x)4^{-k} $$ of $4$-adic digits asymptotic mean of fractional part of…

The article is devoted to the alternating Cantor series. It is proved that any real number belonging to $[a_0-1;a_0]$, where $a_0=\sum^{\infty} _{k=1} {\frac{d_{2k}-1}{d_1d_2...d_{2k}}} $, has no more than two representations by the series…

Let the base $\beta$ be a complex number, $|\beta|>1$, and let $A \subset \C$ be a finite alphabet of digits. The \emph{$A$-spectrum} of $\beta$ is the set $S_{A}(\beta) = \{\sum_{k=0}^n a_k\beta^k \mid n \in \mathbb{N}, \ a_k \in {A}\}$.…

In this paper, we study representations of real numbers in the positional numeration system with negative basis, as introduced by Ito and Sadahiro. We focus on the set $\Z_{-\beta}$ of numbers whose representation uses only non-negative…

For a pair of positive integers $n,k$ with $n\geq 2$, in this paper we prove that $$ \sum_{r=1}^k\sum_{|\bf\alpha|=k}{k\choose\bf\alpha} \zeta(n\bf\alpha)=\zeta(n)^k =\sum^k_{r=1}\sum_{|\bf\alpha|=k}…

For a real number $0<\lambda<2$, we introduce a transformation $T_\lambda$ naturally associated to expansion in $\lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of…