50 problems

Conrey–Farmer–Keating–Rubinstein–Snaith conjecture. For some $\delta>0$,

Universal dominance conjecture. For any modulus $N\ge 3$, the quantity $\log L(1,\chi_{1,a})$ achieves its unique absolute minimum over all non-principal residue classes…

Let $q\not\equiv2\pmod4$, and let $\varphi^*(q)$ denote the number of primitive characters modulo $q$. For each primitive character $\chi$ modulo $q$, write the non-trivial zeros o…

Let $\chi$ be a primitive Dirichlet character modulo $q>1$, and let $L(s,\chi)$ be its Dirichlet $L$-function, with non-trivial zeros $\rho=\tfrac12+i\gamma$. Assume the Generalise…

Let $\chi\pmod{q}$ be a Dirichlet character, and write $s=\sigma+it$. Generalized Lindelöf hypothesis. For every $\varepsilon>0$, if $\sigma\geq 1/2$ and $|s-1|\geq 1/2$, then … Th…

Dirichlet $L$-function moment conjecture. For all $r\in\mathbb{C}$ with $\Re(r)>-3/2$, there is a constant $C_{q,r}$ such that

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…

Pair correlation for Dirichlet $L$-functions. Under GRH, as $x\to\infty$,

Arithmetic exponent pair hypothesis. For every $\epsilon>0$, provided that $|I|<q\delta$, one has

Let $m(R)$ denote the Mahler measure of a non-zero rational function $R$, let $L'(\chi,-n)$ denote the derivative of the Dirichlet $L$-function associated with a character $\chi$,…

Let $m(P)$ denote the Mahler measure of a non-zero polynomial $P$, let $L'(\,\cdot\,,-n)$ denote the derivative of a Dirichlet $L$-function, and let…

Let $d$ range over fundamental discriminants with $|d|\leq X$. Write $\log_2 x=\log\log x$ and $\log_3 x=\log\log\log x$, and let $\theta$ and $\Theta$ be constants satisfying…

Twisted fourth-moment hypothesis. For every coefficient sequence $\boldsymbol{\alpha}$ and every $\varepsilon>0$, the displayed estimate holds. This conjecture asks for the average…

Let $p$ be prime, let $n$ be an integer, and for a Dirichlet character $\chi\bmod p$ define the generalized quadratic Gauss sum … where $e(y)=e^{2\pi i y}$. Let $\chi_0$ denote the…

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 $\sigma\in(\frac12,1)$. Let $q$ be a sufficiently large integer, let $\chi$ be a primitive character modulo $q$, and let $L(s,\chi)$ be its Dirichlet $L$-function. Pointwise up…

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…

Let $k\geqslant1$ be an integer, and let $Q,T\geqslant2$ be real numbers. Let $\mathcal{O}_2(Q)$ denote the family of real Dirichlet characters counted in the paper, and let…

Chowla–Chowla conjecture. $L(2,f)\ne0$, except when

Joint-moment conjecture. As $N\to\infty$,

The logarithmic-derivative conjecture. Then

Bui–Keating moment conjecture. For all non-negative integers $k$,

Let $p$ be a sufficiently large prime, let $m$ be a positive integer, and let $n$ be an integer satisfying $(n,p)=1$. Let $\chi_0$ be the principal character modulo $p$, let…