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We prove simple theorems concerning the maximal order of a large class of multiplicative functions. As an application, we determine the maximal orders of certain functions of the type $\sigma_A(n)= \sum_{d\in A(n)} d$, where A(n) is a…

The generalized Euler number E_{n|k} counts the number of permutations of {1,2,...,n} which have a descent in position m if and only if m is divisible by k. The classical Euler numbers are the special case when k=2. In this paper, we study…

We consider the problem of characterizing all functions $f$ defined on the set of integers modulo $n$ with the property that an average of some $n$th roots of unity determined by $f$ is always an algebraic integer. Examples of such…

Let f(x_1,x_2,...,x_m) = u_1x_1+u_2 x_2+... + u_mx_m be a linear form with positive integer coefficients, and let N_f(k) = min{|f(A)| : A \subseteq Z and |A|=k}. A minimizing k-set for f is a set A such that |A|=k and |f(A)| = N_f(k). A…

Let $r\ge 1$ be an integer and ${\bf U}:=(U_{n})_{n\ge 0} $ be the Lucas sequence given by $U_0=0$, $U_1=1, $ and $U_{n+2}=rU_{n+1}+U_n$, for all $ n\ge 0 $. In this paper, we show that there are no positive integers $r\ge 3,~x\ne 2,~n\ge…

Nicolas inequality we deal can be written as \begin{equation}\label{Nicineq} e^\gamma \log\log N_x < \dfrac{N_x}{\varphi(N_x)}\,, \end{equation} where $x\ge 2$, $N_x$ denotes the product of the primes less or equal than $x$, $\gamma$ is the…

Let $\sigma(n)$ denotes the sum of divisors function of a positive integer $n$. Robin proved that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma}n \log \log n$ holds for every positive integer $n \geq…

We prove that, if f:R^n\to R satisfies Fr\'echet's functional equation and f(x_1,...,x_n) is not an ordinary algebraic polynomial in the variables x_1,...,x_n, then f is unbounded on all non-empty open set U of R^n. Furthermore, the closure…

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive discriminant. In this paper, we consider the Diophantine equation $u_m + u_n = a_1 n_1! + \cdots + a_k n_k!$ and prove that there are only finitely many…

Let $\mathbf{S}$ be the set of all finite or infinite increasing sequences of positive integers. For a sequence $S=\{s(n)\}, n\geq1,$ from $\mathbf{S},$ let us call a positive number $N$ an exponentially $S$-number $(N\in E(S)),$ if all…

We define a family {$\gamma(P)$} of generalized Euler constants indexed by finite sets of primes $P$ and study their distribution. These arise from partial sums of reciprocals of integers not divisible by any prime in $P$. An apparent…

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

For a function $f\colon \mathbb{N}\to\mathbb{N}$, let $$ N^+_f(x)=\{n\leq x: n=k+f(k) \mbox{ for some } k\}. $$ Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and…

In this paper, we prove existence and regularity results for solutions of some nonlinear Dirichlet problems for an elliptic equation defined by a degenerate coercive operator and a singular right hand side. \begin{equation}\label{01}…

It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence.…

A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\mathcal{B},\mu,T)$ and any bounded measurable function $f$, the averages $ \frac1N \sum_{n=1}^N f(T^{s_n}x)$ converge…

Let $a_1 = 1$ and, for $n > 1$, $a_n = a_{n-1} + a_{\left \lfloor \frac{n}{2} \right \rfloor}$. In this paper we will look at congruence properties and the growth rate of this sequence. First we will show that if $x \in \{1, 2, 3, 5, 6, 7…

Given a density $f$ on the non-negative real line, D\"umbgen's algorithm is a routine for finding the (unique) log-convex, non-decreasing function $\hat\phi$ such that $\int\hat\phi(x)f(x)dx=1$ and such that the likelihood…

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erd\H{o}s implying that, for any function $\xi(n)$ tending to infinity with…

The factor complexity ${\mathcal C}_{\mathbf u}$ of a sequence ${\mathbf u} = u_0u_1u_2 \cdots$ over a finite alphabet counts the number of factors of length $n$ occurring in $\mathbf u$, i.e., ${\mathcal C}_{\mathbf u}(n) = \#{\mathcal…