16 problems

Let $\mu$ be a probability measure on $\mathbb{C}$, let $X_1,X_2,\ldots$ be independent and identically distributed with law $\mu$, and define $P_n(z)=\prod_{j=1}^n(z-X_j)$. Let ……

Zero-location conjecture. For every $k\neq 2i^2-2$, $i\geq 2$, the zeros of $G(\mathcal{F}_{L(k\Lambda_0)},j)$ are simple and lie inside the interval $[0,1728]$. This is a numerica…

For $x>0$ and steepness parameter $\kappa>0$, define the smooth divisor-sum indicator by … For each odd prime $p\ge 3$, the asymmetric companion-zero conjecture. For sufficiently l…

For a prime $p$, a companion zero at $p$ is a real zero $x\neq p$ of a smooth prime indicator with $|x-p|<1$. A family $x_p^{(\pm)}(\kappa)$ is a prime-zero approximant if…

Smallest-zero conjecture for $E_{\alpha,2}$. For each $\alpha\in[\alpha_0,2]$, the equation

Clunie–Eremenko–Rossi conjecture. The function $f(z)$ has infinitely many zeros.

Let $\chi$ be a character with conductor $q>1$, and define … Assume $T\ge5/7$. Logarithmic error bound conjecture. The zero-counting function satisfies … This is a speculative expl…

Let $N(T,\chi)$ denote the number of zeros of the Dirichlet $L$-function associated with a character $\chi$ having imaginary part in the relevant height range, and let the conducto…

Arc-zero conjecture. For $k\geq14$, all of the zeros of $\Delta_{k,4}$, $\Delta_{k,6}$, and $\Delta_{k,8}$ in the fundamental domain lie on the arc $|z|=1$.

Boundary-zero conjecture. All the zeros of $\Delta_{k,\ell}$ lying in the standard fundamental domain are on the boundary, $|z|=1$ or $x=\pm1/2$.

Let … where $\widetilde q\approx 0.3092493386$ is the positive threshold described in the source. Sokal's partial-theta problem. Is it true that all zeros of $\Theta_0(z;q)$ remain…

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.…

Let … where $q\in\mathbb{C}$ satisfies $0<|q|\leq 1$. Sokal's conjecture. The entire function $F(z;q)$ can have only simple zeros. The conjecture concerns the multiplicity of the z…

Let $\varrho:=\beta+i\gamma$ range over the zeros under consideration of Dirichlet $L$-functions. Zero-location conjecture. All zeros $\varrho:=\beta+i\gamma$ satisfy … The source…

Rightward correspondence conjecture. For $k\in\mathbb{N}$ there is a one-to-one correspondence between the zeros of $\zeta^{(k)}(s)$ and $\zeta^{(k+1)}(s)$, where the zero of…

Asymptotic periodicity conjecture. For all $\varepsilon>0$ and natural numbers $M$ and $N\geq 2$ there is $K\in\mathbb{N}$ such that for all $k>K$ there exists $s\in\mathbb{C}$ and…