54 problems

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…

Let $J_\nu$ and $J_{\nu+m}$ be Bessel functions of the first kind, with $\nu\in\mathbb{R}$ and $m\in\mathbb{N}$. Bourget's hypothesis. The functions $J_\nu$ and $J_{\nu+m}$ have no…

Shapiro's conjecture. If two exponential sums in ${\cal E}$ have infinitely many zeros in common, they are both multiples of some third entire transcendental exponential sum in…

Let $L(s)$ be an $L$-function whose nontrivial zeros lie on the critical line. The Gaussian Unitary Ensemble (GUE) is the ensemble of large random Hermitian matrices. GUE conjectur…

Let $V(x)=\frac{1}{2}x^2+tx^4$, and let $p_{2j}$ and $p_{2j+1}$ denote the even and odd skew-orthogonal polynomials associated with this potential. Zero-distribution conjecture. Th…

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…

Let $\mu$ be a measure on the unit circle, and consider para-orthogonal polynomials whose zeros include a fixed point $w$ lying in $\operatorname{supp}\mu$. The strong limit points…

Let $f$ be a real Fourier integral with spectral gap $(-a,a)$, meaning that its spectrum avoids this interval. The limit average density of zeros is the limiting average number of…

Let $(\alpha_n)$ be a sequence of non-integers such that … exists, with $0<A<1$. Consider the scaled Laguerre polynomials $L_n^{(\alpha_n)}(nz)$. Martínez et al.'s conjecture. The…

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

Asymptotic zero-locus conjecture. For each $\delta>0$, the zeros of $g_{k,m}$ asymptotically approach the curve $\mathcal{C}_\delta$.

Let $q\in\mathbf{Z}_{\geqslant 1}$, let $A>1$, and let $\mathscr X_q$ be a set of Dirichlet characters modulo $q$. Enumerate in non-decreasing order, with multiplicities, the posit…

For $q\geqslant 3$, consider the multiset of positive imaginary parts of the zeros of all Dirichlet $L$-functions associated with characters modulo $q$. The linear independence con…

Let $\widetilde{\Xi}_n$ be the polynomial defined in the paper, and let its smallest zero be denoted by $z_n$. Smallest-zero conjecture. The smallest zero of $\widetilde{\Xi}_n$ co…

Complex-parameter Laguerre zero distribution conjecture. Then $\llbracket L_n^*(\cdot;b_n,c_n)\rrbracket_n$ converges weakly on $\mathbb{C}$, as $n\to\infty$, to a probability meas…

Complex-parameter Hermite zero distribution conjecture. For every complex $s\in\mathbb{C}$, the probability measures $\llbracket H_n^*(\cdot;s/n)\rrbracket_n$ converge to a probabi…

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…

Zero-distribution conjecture. For sufficiently large $n$, if $n-K$ is even, then $q_n$ has $(n-K)/2$ negative zeros and $(n-K)/2$ positive zeros. If $n-K$ is odd, then $q_n$ has at…

Farmer–Rhoades conjecture. The displayed inequality holds for every $P\in\mathcal{LP}1$ with its zeros listed in increasing order and every pair of consecutive zeros $\xi<\eta$ of…

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 $\alpha,\beta>-1$, $a<0$, and $n\in\mathbb{N}$. For all sufficiently large values of $\gamma$, the negative zeros of $P_n^{(\alpha,\beta,\gamma)}(x;a)$ decrease with respect to…

Let $P_n^{(\alpha,\beta)}$ be the Jacobi polynomial whose zeros satisfy … The parameters $\alpha$ and $\beta$ represent the strengths of repulsion from the endpoints $1$ and $-1$,…

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…

Let $l>-\frac{1}{2}$, and let $z_l(k)$ be the real root of $\mathcal M_{k,l}$ defined near $k=l+\frac{3}{2}$ by $z_l(l+\frac{3}{2})=1+2l$. The conjecture is that its maximal interv…