13 problems

Let $p_t(n)$ denote the number of $t$-multipartitions of $n$. Let $a$ and $t$ be odd integers, with $3\mid t$ whenever $3\mid a$. Define $b$ by … when $3\mid t$, and by…

Let … be an eta-quotient shifted so that all powers of $q$ are integral, and let $\delta_F$ be the odd density of its coefficients $c(n)$, when this density exists. A polynomial is…

Let $m_0$ be the odd part of an even integer $m$, and let $b_m(n)$ denote the coefficients of the corresponding $m$-regular partition generating function. Theorem gives congruences…

Noncongruence and odd-density conjecture. The following assertions hold:

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…

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…

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…

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…

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…