57 problems

Erdős–Granville–Pomerance–Spiro conjecture. If $\mathcal{A}$ has asymptotic density zero, then $s^{-1}(\mathcal{A})$ also has asymptotic density zero.

Folklore conjecture. The number $20$ has no friends; equivalently, $20$ is a solitary number.

Let $\tau_{\ell}(n)$ and $\tau_k(n)$ denote the $\ell$-th and $k$-th divisor functions, respectively, with $\ell,k\in\mathbb{N}$, and let $m$ be fixed. Ng's shifted convolution sum…

Amdeberhan–Andrews–Ballantine conjecture.

Let $G_k$ denote the set of positive integers whose index ratio is $k$. Uniqueness conjecture for subunit index-ratio classes. If $k<1$, then $G_k$ is either empty or has only one…

Let $p$ be a prime number, and let $n$ be a $p$-index ratio number, with divisors $d_1<d_2<\cdots<d_{\tau(n)}$. Prime index-ratio divisor-pair conjecture. Then … Theorem 9 proves t…

Let $n$) be a positive integer with divisors $d_1<d_2<\cdots<d_{\tau(n)}$, and let $k$ be its index ratio, meaning that the sum of the even-indexed divisors equals $k$ times the su…

For integers $k,l,m\ge 2$, define … For $\epsilon>0$ and $0<h\le x^{1-\epsilon}$, the triple convolution conjecture. … as $x\to\infty$, where $\nabla_{h,k,l,m}$ is the explicit Eul…

Let $d_k(n)$ denote the number of ordered factorizations of $n$ into $k$ positive integers, and let $d_l(n)$ be defined similarly. Let $\boldsymbol{\epsilon>0}$ and $k,l\ge 2$. For…

Let $k(m)$ be the maximum number of distinct values of $g_m(n)$ attained at a fixed endpoint in the paper's construction, and let $d(m)$ denote the number of positive divisors of…

Let $k,l\geqslant 2$ be natural numbers, let $h\in\mathbb{Z}\setminus\{0\}$, and let $d_k(n)$ and $d_l(n)$ denote the $k$-fold and $l$-fold divisor functions. Divisor correlation c…

Conjecture 1. If $b\mid\sigma(bk-1)$, then $b\in B$; furthermore, there are no other values of $b$ satisfying this property.

Let $\sigma(n)=\sum_{d\mid n}d$ be the sum-of-divisors function, let $c_{\rm E}$ denote Euler's constant, and let $n$ be a positive integer. Robin's criterion. The Riemann hypothes…

Perfect-number conjecture. There are infinitely many even perfect numbers, but no odd perfect numbers.

Aoudjit–Berkane–Dusart conjecture. The inequality

Let $f_1(n)=\frac{\sigma(n)}{n}$, where $\sigma(n)=\sum_{d\mid n}d$, let $a\geq 1$, and let $\varepsilon>0$. Wolke's conjecture. The inequality … has infinitely many solutions in p…

Matrix convergence conjecture. The first $M-1$ entries of

General constraint conjecture.

Matrix inversion conjecture. The values of the divisor functions $sigma_{a=1,3,5}(n)$ can be obtained by inverting certain matrices, without employing divisibility techniques.

Let $\varphi(n)$ denote Euler's totient function and let … denote the sum-of-divisors function. Erdős's conjecture. The Euler totient function $\varphi(n)$ may be replaced by the s…

Karatsuba's conjecture. As $x\to\infty$,

Let $k$ be the divisor-function parameter, let $w$ be a smooth function supported in $[1,2]$ satisfying … and whose Mellin transform … satisfies rapid decay on every fixed vertical…

Erdős–Kac conjecture. The number $\alpha_k$ is irrational for every positive integer $k$. The conjecture is known unconditionally for $k\leq 3$, and the paper proves it for $k=4$;…

For integers $N$ and real $a\leqslant b$, let $\tau(N;a,b)$ denote the number of divisors of $N$ in $[a,b]$. Ruzsa's conjecture. For every $\varepsilon>0$ there is…

For integers $m,N$ and real $\gamma$, let $\tau(m;a,b)$ denote the number of divisors $d$ of $m$ satisfying $a\leqslant d\leqslant b$. Short-interval divisor conjecture. For every…