11 problems

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 $\theta\in(-1,0]$, let $T$ tend to infinity, and let $\tau$ be uniformly distributed on $[T,2T]$. Write $\mathcal{N}(0,1)$ for a standard Gaussian random variable, and let…

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…

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…

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

Refined Fyodorov–Hiary–Keating conjecture. As a sequence of random variables, the maximum satisfies

The sine-process maximum-growth conjecture at zero shift. The same asymptotic formula holds for $A=0$: