Related papers: On the Euler function of linearly recurrence seque…

For a function $f\colon \mathbb{N}\to\mathbb{N}$, define $N^{\times}_{f}(x)=\#\{n\leq x: n=kf(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…

Translation from the Latin of Euler's "Demonstratio theorematis circa ordinem in summis divisorum observatum" (1760). E244 in the Enestroem index. In his previous paper E243, Euler stated the pentagonal number theorem and assuming it proved…

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12,…

Let $X\subset\mathbb{R}^n$ be a convex closed and semialgebraic set and let $f$ be a polynomial positive on $X$. We prove that there exists an exponent $N\geq 1$, such that for any $\xi\in\mathbb{R}^n$ the function…

Assuming the validity of Dickson's conjecture, we show that the set $\mathcal{V}$ of values of the Euler's totient function $\varphi$ contains arbitrarily large arithmetic progressions with common difference 4. This leads to the question of…

An increasing sequence $(a_n)$ of positive integers which satisfies $\frac{a_{n+1}}{a_n}>1+\eta$ for some positive $\eta$ is called a lacunary sequence. It has been known for over twenty years that every lacunary sequence is strong sweeping…

We extend our result on the convergence of double recurrence Wiener-Wintner averages to the case where we have a polynomial exponent. We will show that there exists a single set of full measure for which the averages \[ \frac{1}{N}…

This paper deals with the existence of positive solutions for the nonlinear system q(t)\phi(p(t)u'_{i}(t)))'+f^{i}(t,\textbf{u})=0,\quad 0<t<1,\quad i=1,2,...,n. This system often arises in the study of positive radial solutions of…

Let $(U_n)_{n=0}^\infty$ and $(V_m)_{m=0}^\infty$ be two linear recurrence sequences. For fixed positive integers $k$ and $\ell$, fixed $k$-tuple $(a_1,\dots,a_k)\in \mathbb{Z}^k$ and fixed $\ell$-tuple $(b_1,\dots,b_\ell)\in…

Let $\phi(n)$ be the Euler-phi function, define $\phi_0(n) = n$ and $\phi_{k+1}(n)=\phi(\phi_{k}(n))$ for all $k\geq 0$. We will determine an asymptotic formula for the set of integers $n$ less than $x$ for which $\phi_k(n)$ is $y$-smooth,…

Let $L(G)$ denote the space of integer-valued length functions on a countable group $G$ endowed with the topology of pointwise convergence. Assuming that $G$ does not satisfy any non-trivial mixed identity, we prove that a generic (in the…

In this paper, we give a short elementary proof of the well known Euler's recurrence formula for the Riemann zeta function at positive even integers and integral representations of the Riemann zeta function at positive integers and at…

Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. The aim of this article is to give a result about the sum of euler's totient function from k equal 1 to n whene p divides n and p…

Let $\vfi$ be Euler's function, $\ga$ be Euler's constant and $N_k$ be the product of the first $k$ primes. In this article, we consider the function $c(n) =(n/\vfi(n)-e^\ga\log\log n)\sqrt{\log n}$. Under Riemann's hypothesis, it is proved…

Let $q$ be a large prime number, $a$ be any integer, $\epsilon$ be a fixed small positive quantity. Friedlander and Shparlinksi \cite{FSh} have shown that there exists a positive integer $n\ll q^{5/2+\epsilon}$ such that $\phi(n)$ falls…

Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ \varphi(n)$ be the Euler totient function. The asymptotic formula for the new finite sum over the primes $ \sum_{p\leq…

Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} \omega_n \frac{z^n}{n!} \] with unimodular coefficients $|\omega_n|=1$. We show that either the counting function $n_F(r)$ of zeroes…

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k…

The $k$-dimensional generalized Euler function $\varphi_k(n)$ is defined to be the number of ordered $k$-tuples $(a_1,a_2,\ldots, a_k) \in \mathbb{N}^k$ with $1\leq a_1,a_2,\ldots, a_k \leq n$ such that both the product $a_1a_2\cdots a_k$…

In this paper, we study some graph theoretical properties of two derivative Euler Phi function set-graphs. For the Euler Phi function $\phi(n)$, $n\in \mathbb{N}$, the set $S_\phi(n) =\{i:\gcd(i,n)=1, 1\leq i \leq n\}$ and the vertex set is…