45 problems

Brück's conjecture. If

Generalized Yang's conjecture. If $f(z)^n f^{(k)}(z)$ is a periodic function, then $f(z)$ is also a periodic function.

Let … where $q$ is a complex number satisfying $0<|q|\leq 1$. Sokal's stronger conjecture. The function $\mathcal F(z;q)$ can have only simple zeros with distinct absolute values.…

Jordan-domain conjecture. Every bounded immediate attracting or parabolic basin of a transcendental entire function is a Jordan domain.

Differentiation is better than midpoint conjecture. The gaps between consecutive zeros of $f'$ lie between the infimum and supremum of the corresponding two-step midpoint-process g…

Let $v$ be a vector field satisfying the assumptions of the Liouville conjecture for stationary Navier–Stokes fields. In addition, suppose that $v$ is the restriction to…

$(Q_n)$ conjecture. $(Q_n)$ is true for all nonnegative integers $n$.

Extremal-constant problem. In the previous problem, $c=q_\infty$.

The remainder characterization conjecture. The sequence ${\mathbf A}$ is a $\mathrm{TP}_\infty$-preserver if and only if, for every $l\in\mathbb N\cup\{0\}$, $R_l[\mathbf A]$ is an…

Weighted Paley–Wiener conjecture. The following equivalences hold:

Zhang et al.'s conjecture. If $f$ and $\Delta^n f$ share $0$ CM, then $f$ has the form

Liu–Laine's conjecture. If $f$ and $\Delta_c^n f$ share $a\in\mathbb{C}\backslash\{0\}$ CM, then

Let $f$ be a non-constant entire function, let $k$ be an integer, and let $a_1$ and $a_2$ be two distinct small functions of $f$. Li and Yang's conjecture. If $f$ and $f^{(k)}$ sha…

Let $q>1$ be fixed and let $p\geq 5$ be an odd prime. Theorem provides a real zero of $\mathfrak F(\cdot,q)$ in $(p-1,p)$. Uniqueness conjecture. For every such $q$ and $p$, this z…

Let $k$ be a positive integer, let $a,b\in\mathbb{C}$ with $a\neq b$, let $a_i\in\mathbb{C}$ for $i=0,1,\ldots,k$ with $a_k=1$, and let $\varphi$ be an entire function. For a funct…

Gao's conjecture. This equation has no entire solution.

Gao's conjecture. If $9ab^2=-4$, then every entire solution is either

Jossen's conjecture. (i) If $f$ and $g$ are two $E$-functions such that $f/g$ is entire, then $f/g$ is an $E$-function. (ii) If two $E$-functions $f$ and $g$ share at least one com…

Fischer operator bijectivity conjecture. The operator $F_P$ is a bijection. The source states that this conjecture remains open in general, although it records several special case…

The $TP_\infty$-preserver conjecture. The sequence ${\mathbf A}$ is a $TP_\infty$-preserver if and only if, for every $l\in\mathbb{N}\cup\{0\}$, the formal power series

Connectedness-locus embedding conjecture. There exists a continuous map from $\mathcal{M}_{\exp}$ to $\mathcal{M}_{\mathrm{err}}$ that is a homeomorphism onto its image.

Smooth-coefficient conjecture. The function $H(x;t)$ satisfies the following properties:

First Hankel relaxation Riemann hypothesis.

$N$-th Hankel relaxation Riemann hypothesis. The function $\Lambda$ is in the $N$-th order Laguerre–Pólya class. These are proposed relaxations of the Riemann hypothesis: membershi…

Let $\phi(x)$ belong to the positive shifted Laguerre–Pólya class $\mathcal{SL\text{-}P}^{+}$, and let $L_j$ be the Laguerre operator defined by the coefficient expansion … The Lag…