39 problems

Let the positive ordinates of the zeros of all principal automorphic $L$-functions be counted with multiplicity. Grand Simplicity hypothesis. This multi-set of positive ordinates i…

Pair correlation conjecture. The normalized form factor is

Let $f$ be a holomorphic cusp form, and consider its zeros in shrinking balls around infinity, in the regime where the height parameter satisfies $H\gg\sqrt{K\log k}$. Ghosh–Sarnak…

Let $F(z)=p(z)+\overline{q(z)}$ be a harmonic polynomial with $p$ and $q$ of degrees $n$ and $m$, respectively, and let $N_F$ denote its number of zeros. Lee–Lerario–Lundberg conje…

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.…

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…

Conjecture on nontrivial-zero trajectories. The limit

Conjecture on trivial-zero trajectories. The limit

For the Riemann zeta function, let $K(\alpha)$ denote the normalized form factor associated with the pair correlation of its nontrivial zeros. Goldston–Montgomery GUE form-factor c…

Let … where $\{t_n^{(m)}\}$ is the sequence of nonnegative zeros of $\mathsf{Z}^{(m)}(t)$, counted with multiplicity. In particular, $\Theta^{(0)}$ measures the limsup of normalize…

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

Equidistribution conjecture for zeros. The zeros of $f$ equidistribute in $\mathcal{F}$ with respect to the hyperbolic measure as $k\rightarrow\infty$. This consequence of the holo…

Let $\tilde F_{\beta}(s)=\sum_{j\ne0}c_j\,\pi^{-(s+i\lambda_j)}\Gamma(s+i\lambda_j)\zeta_{\beta}(s+i\lambda_j)$ be a shifted combination of completed Dirichlet series, with the coe…

Let a linear combination of Dirichlet $L$-functions satisfy natural conditions, notably a suitable notion of independence among the constituent $L$-functions. The Montgomery conjec…

Let $L(s,\chi)$ be a Dirichlet $L$-function, and call a zero non-trivial if it is a zero in the critical strip rather than a trivial zero. Simplicity conjecture. All non-trivial ze…

Let $J{ u}^{(2)}(z;q)$ be the second Jackson $q$-Bessel function, and let ${z{m,nu}}{m=1}^{infty}$ denote its positive zeros. Ismail's conjecture concerns the asymptotic behavior o…

Ford–Zaharescu's conjecture. For every interval $[?]\subseteq\mathbb{T}$,

Positivity conjecture. For all $x\in\mathbb{R}$ and $n\in\mathbb{N}$,

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…

Let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$ act on the upper half-plane, and let … be a standard fundamental domain. For an even integer $k\geq 2$, let … be the normalized Eisenst…

Morris–Feldstein–Bowen conjecture. The consecutive positive zeros satisfy

Let $f_k$ and $\widehat f_k$ be the modified auxiliary functions, and suppose that complex roots may eventually reach the real axis. Vanishing unwanted sign changes conjecture. Eve…