Related papers: Some normal numbers generated by arithmetic functi…
Building upon previous works of Andr{\'e} and Chudnovsky, we prove a general result concerning the approximations of values at rational points a/b of any G-function F with rational Taylor coefficients by fractions of the form n/(B…
The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the least number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We prove that there is a positive constant $c$…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
Given a finite set of real numbers $A$, the generalised golden ratio is the unique real number $\mathcal{G}(A) > 1$ for which we only have trivial unique expansions in smaller bases, and have non-trivial unique expansions in larger bases.…
Let $\gcd(k,j)$ be the greatest common divisor of the integers $k$ and $j$. For any arithmetical function $f$, we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is…
In this article, we present relations for the Euler totient function $\varphi(n)$ and the number of divisors $\tau(n)$ in terms of finite sums of integer parts of rational numbers or greatest common divisors of pairs of integers. Some of…
We construct a uniformly discrete, and even sparse, sequence of real numbers $\Lambda=\{\lambda_n\}$ and a function g in $L^2(R)$, such that for every q>2, every function f in $L^2(R)$ can be approximated with arbitrary small error by a…
Beginning in 2006, G. Gentili and D.C. Struppa developed a theory of regular quaternionic functions with properties that recall classical results in complex analysis. For instance, in each Euclidean ball centered at 0 the set of regular…
We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x(log x)^{-1+o(1)} numbers not exceeding x common to the ranges of Euler's function phi(n) and the sum-of-divisors function sigma(m).
We define an S function as the sum of the asymptotic error terms of digamma function of an arithmetic series, $S(a) \equiv \sum_{n=1}^\infty \left[\ln\frac{n}{a} - \frac{a}{2n}-\psi\left(\frac{n}{a}\right)\right]$, and show a few properties…
A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we are investigating a proper subset of the left-computable numbers. We say that a real number…
In this short note, we give a proof, conditional on the Generalized Riemann Hypothesis, that there exist numbers x which are normal with respect to the continued fraction expansion but not to any base b expansion. This partially answers a…
Let $\alpha$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum $\sum_{n \le x}\alpha(n) f(n)$, normalised to have mean square $1$, has a non-Gaussian…
We say that the order of an algebraic number $A$ is the minimum of positive integers $k$ such that $A^k$ is rational. In this paper, we show that the number of algebraic numbers $A$ with order $k$ such that \[ A,\ A^A,\ A^{A^A},\ \ldots \]…
The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the…
Based on elementary methods and techniques, the explicit formula for the generalized Euler function $\varphi_{e}(n)(e=8,12)$ is given, and then a sufficient and necessary condition for $\varphi_{8}(n)$ or $\varphi_{12}(n)$ to be odd is…
D. Bailey and R. E. Crandall recently formulated a "Hypothesis A", which provides a general principle to explain the (conjectured) normality of constants like pi or log 2 and other related numbers, to base 2 or other integer bases. This…
We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…
The theory of normality for base $g$ expansions of real numbers in $[0,1)$ is rich and well developed. Similar theories have been developed for many other numeration systems, such as the regular continued fraction expansion,…
Let F be a family of holomorphic functions and let K be a constant less than 4. Suppose that for all f in F the second iterate of f does not have fixed points for which the modulus of the multiplier is greater than K. We show that then F is…