20 problems

Let $S(t)$ denote the error term in the number of zeros of the Riemann zeta-function up to height $t$. Limsup conjecture. … The conjecture is motivated by arguments analogous to th…

Diamantis–Rolen odd-part containment conjecture. There is a real number $b=b(f,m,k)>0$ such that

Let $p(x)$ be a real polynomial of even degree $n$. For a real polynomial $q$, write $\sharp_r q$ for the number of its real zeros. Shapiro's Conjecture 12. … The conjecture concer…

For a modular form in the Miller basis, let its associated Faber polynomial be the polynomial whose zeros correspond to zeros of the modular form on the relevant boundary component…

Conjectures on extreme values. One may conjecture that the number of extreme values up to height $T$ is less than $C\log T$ for some constant $C$, and that the sets of indices $n$…

Let $k$ and $\ell$ be fixed natural numbers, and define … Here $B_m$ denotes the $m$th Bernoulli number; the variable $Z$ replaces $z^{2\ell}$ in the one-variable generalization of…

Conjecture on the zeros of $R_{2k+1,p}$. The only real root of $R_{2k+1,p}(z)$ is $z=0$, with multiplicity $2$; moreover, its non-real roots are simple and lie on the circle…

Let $\{P_m(z)\}_{m\geq 0}$ be a sequence of functions of $z$ generated by … Here $a,b,c\in\mathbb{R}\backslash\{0\}$ and $b^2-4ac\leq 0$. Let $t_1$ be the zero of $at^2+bt+c$ havin…

Zero distribution conjecture. The $v$-adic multiple zeta value satisfies

Sign conjecture. On the portion of this curve containing the zeros, the following inequalities hold: if $k$ is even, then

Shapiro's conjecture. Every zero of every $P_n(z)$ that is not a zero of $A(z)$ or $B(z)$ lies on $\mathcal C$. This generalizes Tran's zero-location conjecture to recurrences with…

The maximal-order conjecture for $S(t)$. Conditionally on the Riemann hypothesis,

Zero-location conjecture. All but one of the zeroes of $E_\infty(z)$ in $F_\infty$ lie on the circle

Exceptional-basis conjecture. It is possible to construct such sets $E$ and sequences of random variables such that, for each of these polynomial bases, some subsequence of zero co…

Zero-location and asymptotic conjecture. The set $\mathbb{A}$ is countably infinite and contained in $\mathbb{R}$. Its elements can be enumerated as ${x_n}_{n\geq1}$ with…

Let $\mathit{He}_n(x)$ denote the probabilists' Hermite polynomial of degree $n$, defined by … The preceding properties imply that consecutive Hermite polynomials have no common ze…

Lower-density conjecture. There exists a positive constant $c_{\phi}$ such that, for any $f(z) \in \operatorname{\mathcal{LP}}(r)$ with $r>0$,

Let $varPsi(z)=\operatorname{Max}\{\widetilde H_i(z)\}$, let $K$ be the support of $triangle\varPsi$, and let $mu$ be an asymptotic measure associated with the hypergeometric polyn…

Complex-parameter clustering conjecture. The zeros of $p_n$ asymptotically cluster on the loop of the lemniscate

Let $\Psi$ be the set of nontrivial additive characters of $\mathbb{F}_p$, let $\mathcal{F}_d$ be the family of polynomials under consideration, and let $L_{f,\psi}$ have normalize…