8 problems

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…

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 $q\geq 1$ 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 $q$ 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…

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.

Erdős–Pomerance–Sarkőzy conjecture.