33 problems

General constraint conjecture.

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…

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…

The orthogonal sector variance conjecture. If $x\leq K^\ell$, then there is a constant $a^O_\ell\in\mathbb Q$, depending on $\ell$, such that

Let $d_k(n)$ denote the $k$-fold divisor function, and define … For $x^{1/k}\leq y$ and primes $p$ with $y\leq p\leq 2y$, write $\mathrm{Var}_{p\in[y,2y]}$ for the variance over th…

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

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…

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

Let $T$ be sufficiently large and let $K=T^{1+\eta}$ with $\eta\in(0,1)$. For $k\in\mathbb{N}$, define the $k$-th divisor function by … Define … … where … for $n\in\mathbb{Z}^+$ an…

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…

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…

Fix $k\ge2$. Let $w(y)$ be a smooth function supported in $[1,2]$ such that … and whose Mellin transform $\mathcal{M}[w]$ satisfies … uniformly for $|\sigma|\le A$, for every fixed…

Polynomial divisor-sum conjecture. For every $\epsilon>0$,

Merca's conjectures. The first congruence holds for all $n\in\mathbb{Z}^+$ if and only if $m\in\{5,6\}$; the second holds for all $n\in\mathbb{Z}^+$ if and only if $m\in\{2,3,6\}$;…

Powerfree preservation conjecture. On a set of integers $n$ of asymptotic density $1$,

Let $U_n$ be uniformly distributed on $\{n+1,\ldots,2n\}$, let $d_k(N)$ be the $k$-th divisor function, and set $\Delta_k(U_n)=d_k(U_n)-\mathbb{E}(d_k(U_n))$ and…

Let $a(n)$ denote the recursive divisor function, and call a positive integer $n$ ample when $a(n)>n$. Let $p_1,p_2,\ldots$ be the primes in increasing order. Fink's conjecture. Fo…

Universal odd-prime valuation conjecture. Every $n\in\mathbb{N}$ satisfies condition (1) or (2). Consequently,

Let $d_k(n)$ denote the $k$-fold divisor function, and define the additive divisor correlation $D_{h,k,l}(x)$ as in the paper. For fixed $k,l\in\mathbb{N}$ and…

Let $d_z(n)$ be the generalized divisor function and define the arithmetic-progression variance by … Let $a_z$ and $\alpha_1^{(z)}$ be as in the paper. The arithmetic-progression d…

For $z>0$, let $d_z(n)$ be the generalized divisor function, define $D_z(x)=\sum_{n\leq x}d_z(n)$, and let $\overline D_z(x)$ be the contour approximation defined in the paper. Set…