26 problems

Erdős–Mirsky conjecture. The set $S$ is dense in $\mathbb{R}$.

Keating–Rodgers–Roditty-Gershon–Rudnick's variance conjecture. When $Q^{1+\varepsilon}<X<Q^{k-\varepsilon}$ and $X\to\infty$,

Let $n$ be square-free, let $d(n)$ be the divisor function, and let $\omega(n)$ be the number of distinct prime divisors of $n$. Square-free divisor-function conjecture. There is a…

Browning's conjecture. Based on algebraic and geometric considerations, this sum is asymptotic to $c_hx(\log x)^3$ for a suitable constant $c_h\neq 0$ as $x\to\infty$.

Let $N$ be an index of a highly-composite Mersenne number if $\tau(2^N-1)>\tau(2^m-1)$ for every integer $m$ with $1\leqslant m<N$. Highly-composite Mersenne index conjecture. If…

Let $f(n)=\sum_{1\leqslant k\leqslant n}\tau(2^k-1)$, where $\tau(m)$ denotes the number of positive divisors of $m$. Ratio conjecture. The ratio $f(2n)/f(n)$ should stabilize and…

Erdős's conjecture. The sequence of ratios $d(n+1)/d(n)$ is everywhere dense in $(0,\infty)$.

Let $E/Q$ be a non-CM elliptic curve, and let $\prod_{\rm red}(x)$ denote the primes $p\leq x$ at which $E$ has good reduction. Write $d(n)$ for the divisor function and let $X$ ra…

Kuperberg–Lalin's Gaussian-direction variance conjecture. If $x\leq K^\ell$, then as $x\to\infty$,

Kuperberg–Lalin's symplectic prime variance conjecture. For $x^{1/k+\varepsilon}\leq y$, one has

Let $k$ be a positive integer, let $r$ be a positive fundamental discriminant, and let $\chi_r$ be the primitive quadratic character modulo $r$. Define … For $y>0$, define the vari…

Let $e_q(x)=e^{2\pi i x/q}$, let $\overline{m}$ denote the multiplicative inverse of $m$ modulo $q$, and let $\mathfrak{a}\in\{0,1\}$. For $y>0$, define … where $\int_{(b)}$ denote…

Let $q\geq 1$ be a composite number, and let $W(x)$ be a smooth test weight supported on $[1,2]$. Its Mellin transform is … Assume that, for every fixed $A>0$ and every positive in…

Let $d(x)$ denote the number of positive divisors of $x$. Conjecture on three consecutive integers with four divisors. There are infinitely many positive integers $x$ such that … T…

Let $q$ tend to infinity through primes, let $\mathcal D_k(\alpha)$ be the normalized variance of the $k$-fold divisor function in reduced residue classes, and let…

Let $\mathcal M_k(\alpha)$ be the normalized long-Dirichlet-polynomial mean square and let $\mathcal C_k(\alpha)$ be the normalized divisor variance in short intervals. Basor–Ge–Ru…

For $\alpha>0$ and $k\in\mathbb N$, define $\mathcal C_k(\alpha)$ as the normalized limiting variance of the $k$-fold divisor function in short intervals, and let…

Highly composite minima conjecture. All the integers $n_k$ produced by Theorem 1 are highly composite numbers.

Triple-divisor correlation conjecture. For a suitable constant $c_h>0$,

Let $a$ and $q$ satisfy $(a,q)=1$, and let $x>q^{\theta}$ with $\theta>1$. Write $\tau(n)$ for the divisor function and $\phi(q)$ for the Euler totient function. Divisor-function e…

Ramachandra–Sankaranarayanan conjecture. Assuming the RH, we have

Hall's conjecture. The same asymptotic formula should remain true whenever

Let $d^{(1)}(n)=d(n)$ and, for fixed $ke1$, let $d^{(k)}(n)=d(d^{(k-1)}(n))$. Let $log_k$ denote the $k$ times iterated natural logarithm, and let $c_k$ be a positive constant. Bel…

Averaged additive divisor conjecture. For every fixed $k\geqslant 3$, there should exist some $\delta_k>0$ such that, uniformly for

Vinogradov's conjecture. For the relevant, but unspecified, range of $h$, one should have