25 problems

Hayman's conjecture. The differential polynomial $f^n f'$ cannot have any non-zero generalized Picard exceptional values.

For a meromorphic function $f$ and meromorphic functions of slow growth serving as moving targets, the relevant notions of value distribution are understood in the setting of Nevan…

Denjoy's conjecture. The number of finite asymptotic values of $f$ is at most $2k$.

Growth equality conjecture. The estimates labeled … are equalities, and $\eta=2$ in the case $D_g=3$ and $\eta=1$ in the case $D_g=4$, which implies $\eta=R$ in these cases.

Totally ramified value-number conjecture. For algebraic minimal surfaces, there exists $2.5\leq\kappa<3$ such that $\nu_g\leq\kappa$.

Sharp upper bound conjecture. The sharp upper bound of $D_g$ is $2$.

Let $a_n(k)$ denote the coefficient of $x^k$ in the cyclotomic polynomial of order $n$, let $\delta(a_n(k)=v)$ denote the density of integers $n$ for which this coefficient equals…

Let $r$ be the spectral parameter of a Maass cusp form, with eigenvalue … Let $F$ be any compact regular subregion of the fundamental domain $\mathcal F$, and let $\sigma$ denote t…

Reformulated Brück conjecture. The function $\alpha$ is constant, say $\alpha=d$, and

Let $a\in\mathbb C\setminus\{0\}$. An $a$-value of $\zeta(s)$ is a solution of $\zeta(s)=a$. Let $N_{a,0}(T)$ count the $a$-values on the critical line up to height $T$, and let…

Let $f$ be a meromorphic function, let $a\in\mathbf{C}$ satisfy $a\neq 1$ and $a\neq (n+1)/n$ for every $n\in\mathbf{N}$, and consider the assertion that $ff”-a(f')^2$ has infinite…

Let $f$ be a meromorphic function, and let $\overline{N}(r,\infty,f)$ denote the averaged counting function of distinct poles, counted without multiplicity. Goldberg's conjecture.…

Let $S\subset\mathbb{C}$ be Borel, and let $\zeta(s,\alpha)$ denote the Hurwitz zeta function for a shift parameter $0<\alpha<1$. Gaussian value-distribution conjecture. For large…

Let $J\geq2$, and let $f_j$, $1\leq j\leq J$, be pairwise orthogonal real-valued Hecke–Maass cusp forms for $\operatorname{SL}_2(\mathbb{Z})$, with spectral parameters $t_{f_j}$. A…

Gaussian moments conjecture. For every $n\in\mathbb{Z}_{\geq1}$,

Symmetry conjecture. For $x>0$ and sufficiently small $\delta>0$,

Generalized Shapiro's conjecture. If $f$ and $g$ are not multiples of the same transcendental exponential polynomial of order $q$, then

Radziwiłł's Gaussian value-distribution conjecture. If $V=o(\sqrt{\log\log T})$ as $T\to+\infty$, then

Steuding's conjecture. The value-distribution conclusion above should hold for all $L$-functions with such a polynomial Euler product.

Let $V(\sigma):=\{\zeta(\sigma+it):t\in\mathbb{R}\}$ be the value set of the Riemann zeta-function on the vertical line $\operatorname{Re}(s)=\sigma$. The critical line is…

Let $a$ be a fixed complex number. An $a$-value is a solution $s$ of $\zeta(s)=a$, and it is simple when $\zeta'(s)\neq 0$. Simplicity conjecture. For any fixed complex number $a$,…

Let $\zeta(s)$ be the Riemann zeta-function, and let the critical line be $s=\frac{1}{2}+it$ with $t\in\mathbb{R}$. Density conjecture. The set of values … is dense in the complex…

Zero-existence conjecture. Every function of this form has zeros. The source states this as the main conjecture for these meromorphic functions and gives no resolution.

Let $f$ be a transcendental meromorphic function in the plane and let $k\geq 2$. Write $\overline{N}(r,f)$ for the reduced integrated counting function of poles of $f$,…

Let $f$ be a transcendental meromorphic function in the plane, and let $k\geq 1$. WYY's conjecture. The function $f f^{(k)}$ takes every finite nonzero value infinitely often. The…