20 problems

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…

Differentiation conjecture. There exist sequences $A_n$, $B_n$, $C_n$, and $D_n$ with $D_n/C_n$ bounded, such that

Let $f_n(x)=a_0+a_1x+\cdots+a_nx^n$ be a random polynomial with i.i.d. coefficients satisfying the hypotheses at the beginning of the section, and let $E_n$ denote its expected num…

Let $f_n(x)=a_0+a_1x+\cdots+a_nx^n$ be a random polynomial with nondegenerate, zero-mean, i.i.d. coefficients, and let $p_{n,k}$ denote the probability that $f_{n+1}$ has exactly…

Let $N_{\mathbb{R},\boldsymbol{\xi}}$ be the number of real zeros of a random polynomial from any model considered in Section. Assume that the coefficient distribution is subgaussi…

Let $N_{I,\boldsymbol{\xi}}$ denote the number of real zeros in a natural interval $I$ for one of the random polynomial models considered in the paper. Suppose that the coefficient…

Let $d53d_d(d70e_1,d70e_2)$ be the number of real zeros of $L'(s,\chi_d)$ on the interval $[d70e_1,d70e_2]$, where $\chi_d$ is the primitive quadratic character attached to the fun…

Let $f$ be a real transcendental meromorphic function in the plane with at least one pole, meaning that $f(\mathbb R)\subseteq\mathbb R\cup\{\infty\}$. Assume that all zeros and po…

Hellerstein–Sheil-Small–Wittich conjecture. Then $f$ satisfies

Let $r\geq 1$, and let an inter-trivial zero (ITZ) of $\zeta_r(s)$ mean a real zero on the negative real line other than the known zeros at the negative even integers. For any…

Let $r\geq 2$. For $2\leq k\leq r$, let $I_r(k)$ denote the number of inter-admissible zeros of $\zeta_r(s)$ in the interval $(1/k,1/(k-1))$, where an inter-admissible zero is a re…

Let $r\geq 1$ and $n\in\mathbb{N}$. An inter-trivial zero (ITZ) is a real zero of $\zeta_r(s)$ lying between consecutive trivial zeros on the negative real axis. Inter-trivial zero…

Let $r\geq 2$. For $2\leq k\leq r$, let $I_r(k)$ denote the number of inter-argument zeros (IAZs) of $\zeta_r(s)$ in the interval $(1/k,1/(k-1))$. Simple IAZ conjecture. All IAZs o…

Let $A\subseteq\mathbb{N}$ with $|A|=n$, and define the cosine polynomial … Let $Z(f_A)$ denote the number of zeros of $f_A$ in $[0,2\pi]$. Borwein–Erdélyi–Ferguson–Lockhart conjec…

Let … be a random algebraic polynomial whose coefficients $\{\eta_j\}$ are independent and identically distributed uniformly on $[-1,1]$, and let $N_n(\mathbb{R})$ denote the numbe…

Stable-domain conjecture. Coefficients from a stable domain of attraction should lead to a different limiting mean density function.

Borwein–Erdős–Ferguson–Lockhart conjecture. The number of zeros of $f_A$ tends to infinity as $N$ tends to infinity.

Littlewood's conjecture. The number of zeros of $f_A$ is possibly $N-1$ or not much less.

Let $f$ and $\widehat f$ be the limiting functions from the convergence conjecture, and call the prescribed zeros of the construction the forced roots. No-unwanted-roots conjecture…

Let $D_n(x)$ be the polynomial whose leftmost zero is denoted by $z_n^*$, and let $c_{\mathrm{even}}$ and $c_{\mathrm{odd}}$ be the constants appearing in the asymptotic formulas b…