Related papers: The distribution functions of $\sigma(n)/n$ and $n…

Let $\phi(n)$ be the Euler totient function and $\sigma(n)$ denote the sum of divisors of $n$. In this note, we obtain explicit upper bounds on the number of positive integers $n\leq x$ such that $\phi(\sigma(n)) > cn$ for any $c>0$. This…

Let phi(n) be Euler's totient function and let sigma(n) be the sum of the positive divisors of n. We show that most phi-values (integers in the range of phi) are not sigma-values and vice versa.

Let $\phi(\cdot)$ and $\sigma(\cdot)$ denote the Euler function and the sum of divisors function, respectively. In this paper, we give a lower bound for the number of positive integers $m\le x$ for which the equation $m=n-\phi(n)$ has no…

We obtain an upper bound for the sum $\sum_{n\leq N} (a_{n}/\varphi (a_{n}))^{s}$, where $\varphi$ is Euler's totient function, $s\in \mathbb{N}$, and $a_{1},\ldots, a_{N}$ are positive integers (not necessarily distinct) with some…

We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $\phi(n)$ and the sum-of-proper-divisors function $s(n):=\sigma(n)-n$. We prove that the values $\phi(n)$, for $n\le x$, that are coprime to $p$…

Euler totient function $\phi(n)$ plays a central role in number theory and is applied in areas such as cryptography. In this paper, we study iterations of the totient function. We first prove that for any integer $n>2$, iteratively applying…

We show that for some $k\le 3570$ and all $k$ with $442720643463713815200|k$, the equation $\phi(n)=\phi(n+k)$ has infinitely many solutions $n$, where $\phi$ is Euler's totient function. We also show that for a positive proportion of all…

We show that the equation phi(a)=\sigma(b) has infinitely many solutions, where phi is Euler's totient function and sigma is the sum-of-divisors function. This proves a 50-year old conjecture of Erdos. Moreover, we show that there are…

Paul Erdos and Carl Pomerance have proofs on an asymptotic upper bound on the number of preimages of Euler's totient function $\phi$ and the sum-of-divisors functions $\sigma$. In this paper, we will extend the upper bound to the number of…

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 result $ \sum_{n\leq x}\varphi([x/n])=(6/\pi^2)x\log x+O\left ( x(\log x)^{2/3}(\log\log…

Let $\phi(n)$denote Euler's phi function. We study the distribution of the numbers $gcd(n,\phi(n))$ and their divisors. Our results generalize previous results of Erd\H{o}s and Pollack.

The Euler's totient function $ \varphi(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ \varphi(n)\mid n-1…

Inspired by Lehmer's and Deaconescu's conjectures, as well as various analogue problems concerning Euler's totient function $\varphi(n)$, Schemmel's totient function $S_{2}(n)$, Jordan totient function $J_k$, and the unitary totient…

In this paper, we show that if $(U_n)_{n\ge 1}$ is any nondegenerate linearly recurrent sequence of integers whose general term is up to sign not a polynomial in $n$, then the inequality $\phi(|U_n|)\ge |U_{\phi(n)}|$ holds on a set of…

We consider the distribution of $\arg\zeta(\sigma+it)$ on fixed lines $\sigma > \frac12$, and in particular the density \[d(\sigma) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: |\arg\zeta(\sigma+it)| > \pi/2\}|\,,\] and the…

Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ its $k$-fold iterate. In this note, we improve the upper bound for the number of positive $n\leqslant x$ such that $\phi_{k+1}(n)\geqslant cn$. Comparing with the upper bound which…

Let $\varphi(n)$ denote the Euler totient function. In this paper, we first establish a new upper bound for $n/\varphi(n)$ involving $K(n)$, the function that counts the number of primorials not exceeding $n$. In particular, this leads to…

For any positive integer $n$, let $\sigma (n)$ be the sum of all positive divisors of $n.$ In this paper, it is proved that the set of positive integers $ n $ for which $ \sigma(30n+1)\geq \sigma(30n) $ has a density less than $ 0.0371813,…

The divisor function $\sigma(n)$ denotes the sum of the divisors of the positive integer $n$. For a prime $p$ and $m \in \mathbb{N}$, the $p$-adic valuation of $m$ is the highest power of $p$ which divides $m$. Formulas for…

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…