33 problems

Let $D$ be the discriminant parameter for the imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$, and let $\chi$ range over class group characters of $\mathbb{Q}(\sqrt{-D})$. Write…

Let $\zeta(s)$ denote the Riemann zeta function, let $\gamma$ be Euler's constant, and let $C_1$ be a constant. Write $\log_2 T=\log(\log T)$ and $\log_3 T=\log(\log(\log T))$. Ref…

Let $\zeta(s)$ denote the Riemann zeta function, let $\gamma$ be Euler's constant, and write $\log_1 T=\log T$ and $\log_j T=\log(\log_{j-1}T)$ for $j\geq 2$. The notation…

Extreme-value conjecture. Given the extraordinary decay of $\Phi(\tau)$ in the relevant range, the above estimate should hold, at worst with a slightly different constant.

Let $T\to\infty$ and consider the maximum of $|\zeta(1/2+it)|$ for $T<t<2T$. The source records an upper bound under the Riemann Hypothesis and a lower bound attained along a seque…

Let $P_N(z)$ be the characteristic polynomial of an $N\times N$ random matrix, and let $\mathcal{I}$ be a fixed macroscopic interval in the bulk of the spectrum or a fixed finite a…

Joint-tail conjecture. As $u\to\infty$,

Let $L(s,\chi)$ be the Dirichlet $L$-function, and let the maximum range over primitive nontrivial Dirichlet characters $\chi$ modulo integers $q\leq Q$. Lamzouri's maximum-order c…

Let $q>1$, let $\mathcal{X}$ be the group of Dirichlet characters modulo $q$, and let $H\leq\mathcal{X}$ be a subgroup satisfying $\#H\gg q^\varepsilon$. For any coset representati…

Let $q>1$, let $\mathcal{X}$ be the group of Dirichlet characters modulo $q$, and let $L(\chi,s)$ denote the associated Dirichlet $L$-function. Gonek-Hughes-Farmer conjecture. As…

Let $0<\alpha<1$ and set … Let $\tau$ be uniformly distributed on $[T,2T]$, and let $(\mathcal{M}_T,T>1)$ be a sequence of random variables. Define $\beta_c=2\sqrt{1+\theta}$. Hybr…

Arguin, Dubach, and Hartung's conjecture concerns the maximum of a random model of the Riemann zeta function over intervals whose lengths vary between the IID and log-correlated re…

Maximal-order conjecture.

Let $q$ tend to infinity through the primes. For a prime $q$, let $\chi_0$ be the principal character modulo $q$, and let $L(s,\chi)$ be the Dirichlet $L$-function associated with…

Let $\zeta(s)$ denote the Riemann zeta function, and write $\log_2 T=\log\log T$ and $\log_3 T=\log\log\log T$. Maximal logarithmic derivative conjecture at $1$. As $T\to\infty$, ……

Fix $\epsilon\in(0,\frac12)$. For $T>0$, consider the interval $[T,2T]$, and let $\zeta(s)$ denote the Riemann zeta function. Directional maximum conjecture. As $T\to\infty$, … uni…

Let $q$ tend to infinity through the primes. For a prime $q$, let $\chi_0$ denote the principal character modulo $q$, and let $L(s,\chi)$ be the Dirichlet $L$-function associated w…

Extreme-value conjecture. The largest values of $\log|Z(t)|$ are conjectured to be

Let $cmathcal F$ be a reasonable family of $L$-functions for $cmathrm{GL}_n$, let $C(L)$ denote the analytic conductor, and normalize the $L$-functions so that their central value…

Derivative extreme-value conjecture. As $T\to\infty$,

Let $\ell\in\mathbb N$ and $\sigma\in(0,1)$ be fixed. Comparison conjecture. There exist constants $C(\sigma,\ell)$ and $c(\sigma,\ell)$, depending on $\sigma$ and $\ell$, such tha…

Let $\ell\in\mathbb N$ be fixed, and write $\log_2 T=\log\log T$ and $\log_3 T=\log\log\log T$. Polynomial asymptotic conjecture. There exists a polynomial $P_{\ell+1}(x,y)$ of tot…

Let $T$ be sufficiently large, and for a positive integer $\ell$ write $\zeta^{(\ell)}$ for the $\ell$th derivative of the Riemann zeta function. Monotonicity conjecture. Uniformly…

Let $\gamma$ denote the Euler–Mascheroni constant, and write $\log_2 T=\log\log T$ and $\log_3 T=\log\log\log T$. Littlewood's conjecture. … This conjecture concerns the maximal si…

The critical-threshold conjecture. The supremum is attained in the limit $t,s\to0$ with $t\ne s$, and