88 problems

Let $n$ be a positive integer, let $\phi$ be the classical Euler totient function, and let $\phi^{-1}(n)$ denote its inverse image. Carmichael's conjecture. For every positive inte…

Let $g_\ell(n)$ denote the number of subgroups of index $n$ in $\mathbb{Z}^\ell$, and write … Call a sequence $(a_k)_{k\in\mathbb{N}}$ log-concave at $k\geq 1$ when…

For a positive integer $n$, let $\omega(n)$ be the number of distinct prime factors of $n$. For $x>0$, count integers $n\leq x$ satisfying $\omega(n)=\omega(n+1)$. Erdős–Pomerance–…

Let $\varphi(n)$, $\psi(n)$ and $\sigma(n)$ denote the Euler totient, Dedekind psi and sum-of-divisors functions, respectively. For integers $k\geq1$ and $n\geq2$, use…

Let $S_2(n)$ denote Schemmel's totient function, defined on prime powers by … where $p$ is prime and $k\in\mathbb{Z}^{+}$. Deaconescu's conjecture. For every integer $n\geq 2$, … i…

Erdős's conjecture. The counting function satisfies

Fixed-point divisor-sum average-order conjecture. The average order of

For a given arithmetic function $f$, let $F_f(x)$ be the largest $k$ for which there exist integers $n$ with $n+k\leq x$ and … Here $\varphi$ denotes Euler's totient function. Erdő…

Let … where the product runs over all prime powers unitarily dividing $N$; this is the unitary totient of $N$. Subbarao's conjecture. … if and only if $N$ is a prime power. This is…

Let $\theta$ be the supremum of real numbers $c$ such that there are $\gg x/\log x$ primes $p\leq x$ for which $p-1$ has a prime factor greater than $p^c$. Shifted-prime exponent c…

Blomer and Granville's conjecture. As $x$ tends to infinity,

Auxiliary solution-count conjectures.

Let $p$ and $q$ be primes, and let $x$ be a real number. Alaoglu–Erdős conjecture. If both $p^x$ and $q^x$ are rational, then $x$ is an integer. The question was raised in connecti…

Let $f$ be any small arithmetic function, and let $S_f(x)$ and $C_f$ be as in the paper. The $2/5$ error-term conjecture. One has … The preceding mean-square theorem shows that the…

Let $\varphi(n)$, $\psi(n)$ and $\sigma(n)$ denote the Euler totient, Dedekind psi and sum-of-divisors functions, respectively. For integers $k\geq1$ and $n\geq2$, form the three q…

Let $\varphi(n)$, $\psi(n)$ and $\sigma(n)$ denote the Euler totient, Dedekind psi and sum-of-divisors functions, respectively. For integers $k\geq1$ and $n\geq2$, consider the thr…

McCranie's conjecture. Every composite integer $n$ satisfying

For a nonnegative integer $s$, define … Here $\varphi_k(n)$ is the generalized totient function … and $\varphi_0(n)=\varphi(n)$ is Euler's totient function. Büyükaşık–Göral–Sertbaş…

Let $k$ be a positive integer, let $n$ range over positive integers, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. Generalized Erdős–Sierpiński conjecture. Fo…

The lucky divisibility functions are defined by … where $\overset{*}{\omega}(n)$ counts distinct lucky divisors and $\overset{*}{\Omega}(n)$ counts them with their lucky order. Luc…

Let $\varphi^+(n)$ denote the unitary totient function of the positive integer $n$. The conjecture on $\varphi^+$ values. For each prime $q\geq 3$, there are infinitely many pairs…

Let $\varphi(n)$ denote Euler's totient function. Conjecture on totient increases at primes. For infinitely many primes $p$, … The source explicitly says that this conjecture is st…

Let $\sigma^+(n)$ denote the sum of the unitary positive divisors of the positive integer $n$. The conjecture on $\sigma^+$ values. For each prime $q\geq 3$, there exist infinitely…

Iterated totient sum conjecture.

Let $e(n)=\sigma(n)-2n$ be the excedent function, where $\sigma(n)$ denotes the sum of the positive divisors of $n$. Davis–Klyve–Kraght's conjecture. Every even integer is in the r…