28 problems

Let $c(n)$ denote the coefficient sequence under consideration, and let $n\in\mathbb{N}_0$. The congruence conjecture for $c(32n+23)$. … This conjecture is proposed as an open ques…

Let $N(r,2;n)$ denote the number of partitions of $n$ whose rank is congruent to $r$ modulo $2$, for $r\in\{0,1\}$. Hou–Jagadeesan's conjecture. If $r=0$ (respectively, $r=1$), the…

Let $\textup{spt}_\omega(n)$ denote the partition statistic used in the paper. For integers $k\geq 1$ and $n\geq 0$, Wang's conjecture. … … The conjecture concerns an infinite fami…

The finite sign law. For every $n\geq 0$,

Let $c(n)$ denote the coefficient sequence under consideration, and let $k,n\in\mathbb{N}_0$. The infinite-family congruence conjecture for $c(n)$. … … … The authors propose these…

Andrews's conjecture. The asymptotic series converges conditionally to $\alpha_f(n)$.

Let $p$ be an odd integer of the form $p=6k\pm1$, with $k\in\mathbb Z$, and let $K_p^{(j)}(t)$ and $L_p^{(j)}(t)$ be the linear combinations of Mordell–Borel integrals defined in t…

Half-oddness conjecture. When $n\equiv 7\pmod{8}$, “half” of the values of $c_1(n)$ are odd.

Let … where $(a;q)_n$ denotes the $q$-Pochhammer symbol and $J_a$ and $J_{a,b}$ denote the theta-product notation used for these functions. The fifth-order mock theta conjecture fo…

Mod 4 congruence conjecture. For every $n\geq0$ and every integer $j$ with $0\leq j\leq\ell-2$,

Let $m,m'\in\mathbf N$ and $j,j'\in\mathbf Z$. For $m\in\mathbf N$ and $j\in\mathbf Z$, let $\overset{N=3}{CH}{}^{(+)[K(m),j]}$ denote the $\mathbf C((q^{\frac12}))$-linear span of…

Let $f(q)$ denote any one of the mock theta functions covered by the paper's theorems and corollaries. For an odd positive integer $n$, let … , we have … The paper proves nonvanish…

Let $N_A(n)$ be defined by … Let $p\not\equiv7\pmod8$ be an odd prime, let $\delta_p$ satisfy … and let $k,n\in\mathbb Z$ with $\left(\frac{k}{p}\right)=1$. Kim–Lim–Lovejoy's…

Let $\operatorname{spt}(n)$ denote the number of occurrences of the smallest part among the partitions of $n$. Let $\ell>3$ be prime with $\ell\not\equiv23\pmod{24}$, and define ……

Let $v(n)$ count odd-balanced unimodal sequences of size $2n+2$, and let $v(m,n)$ count those with rank $m$. Let $\ell\not\equiv-1\pmod8$ be prime and let $k$ be a positive integer…

Bryson–Ono–Pitman–Rhoades conjecture. For all $n$,

Let $p_{\xi}(n)$ denote the coefficients of the Gordon–McIntosh mock theta function $\xi(q)$, so that $\xi(q)=\sum_{n\geq 0}p_{\xi}(n)q^n$. Generating-function congruence. One has…

Let $g(q)$ be a mock theta function with integer coefficients, and let $c(g;n)$ denote the coefficient of $q^n$ in its series expansion. Say that $g$ is of parity type…

Let $\overline{\textup{spt}}_\omega(n)$ denote the overpartition smallest-parts function used in the paper. For integers $\ell\geq 1$, $k\geq 0$ and $n\geq 0$, the generalized cong…

Oddness conjecture. The function $Y(q)$ is an odd function of $q$. This conjecture would provide another proof of the mod $4$ congruences in Theorem; the authors state it as an ope…

Let $L(\Lambda)$ be an integrable nonzero-level module over the affine Lie superalgebra in the surrounding section, with $\Lambda\in\Omega_k$ or $\Lambda\in\Omega'_k$. Define…

Mock theta identity. The source records the following identity as a conjectural formula:

Radial-limit identity. The following identity should hold:

For fixed integers $a,b,A,B$, let $f_{a,b,A,B,h,k}$ be the modular forms used to cancel the singularities of $g_2(\zeta_b^a q^A;q^B)$ as $h/k$ ranges over $\mathbb{Q}$. Rhoades' fi…

Mock theta conjecture. The function $f_0(q)$ satisfies