21 problems

Bringmann–Harrison–Heim–Kane's conjecture. The coefficients satisfy

The second author's conjecture. The two vanishing sets are equal:

Cooper's conjecture. The sequences $\{T_{14,\pm}(n)\}$ satisfy the Lucas congruence modulo $p$ if and only if $p=2$ or $p\equiv1,7\pmod{8}$, and the sequences…

Completeness conjecture for dual pairs. The set of $203$ dual pairs of eta quotients of weight $(0,2)$ is the complete set of such dual pairs for all levels $N$.

Odd density conjecture for $b_{21}$. The series

Main parity conjecture for eta-quotients. The following assertions hold: (i) for every $F$, $\delta_F$ exists and satisfies $\delta_F\le 1/2$; (ii) if $\delta_F=1/2$, then…

Noncongruence conjecture for $m=15$ and $m=27$. There is no even arithmetic progression for either $m=15$ or $m=27$.

Odd density conjecture. For every odd integer $t\ge 1$, the coefficients of $p_t(n)$ have odd density $1/2$.

Let the eta-quotients obtained from Theorems and be the eta-quotients constructed in those results. Linear independence conjecture. All eta-quotients gained from Theorems and are l…

Let $N>1$ be an integer, and let $R_{N,1}(n)$ denote the Rademacher-sum sequence associated with level $N$. The genus $g_0(N)$ is the genus of the modular curve corresponding to…

For a prime $p$ and an integer $n>3$, let $f_{p,n}$ be the eta quotient defined by … Here $\eta_m$ denotes $\eta(mz)$, and a simple holomorphic eta quotient is one that is both pri…

Birationality conjecture. This map is birational equivalence for all functions $\Delta_{N,12}$ from Theorem.

Call an eta quotient primitive if it is not a rescaling of another eta quotient by a positive integer. For an eta quotient $f$, its extract is the primitive eta quotient $f_0$ of w…

Let $f$ be an irreducible holomorphic eta quotient, and let an Atkin–Lehner involution act on eta quotients in the usual way. Atkin–Lehner irreducibility conjecture. The images of…

Let $f$ be an irreducible holomorphic eta quotient, and for a positive integer $\nu$ let its rescaling be $f_\nu(z)=f(\nu z)$. Rescaling irreducibility conjecture. The rescalings o…

Let $f$ be a holomorphic eta quotient of level $N$. It is quasi-irreducible if it is not factorizable on $\Gamma_0(N)$, whereas it is irreducible if its only factors are $1$ and…

Let $f$ be a holomorphic eta quotient of level $N$. An eta quotient on $\Gamma_0(M)$ is one whose level divides $M$; $f$ is factorizable on $\Gamma_0(M)$ if there are nonconstant h…

Zagier's finiteness conjecture. For every fixed weight, there are only finitely many simple holomorphic eta quotients of that weight.

A holomorphic eta quotient is simple if it is both primitive and quasi-irreducible: it cannot be obtained from another eta quotient by replacing $z$ with $u z$ for any…

For a prime $p$, let $k_{\max}(p^n)/2$ denote the maximum weight of holomorphic eta quotients of level $p^n$ that are not factorizable on $\Gamma_0(p^n)$, and let $r_n\in\{0,1\}$ b…

A holomorphic eta quotient is quasi-irreducible if it satisfies the paper's definition of quasi-irreducibility, and it is irreducible if it cannot be written as a nontrivial produc…