23 problems

Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathrm{GL}_n$ over $\mathbb Q$, with associated $L$-function … Let…

Let $\mathcal{H}_{2g+1}$ be the set of monic square-free polynomials of degree $2g+1$ over $\mathbb{F}_q[x]$, and define … Let $P_X(\chi_D)=P_X(\tfrac12,\chi_D)$ and…

Loughran–Santens conjecture. For any balanced counting function $C$, the leading constant in Malle's conjecture is equal to a finite sum of Euler products indexed by certain Brauer…

Kaczorowski's degree-equals-degree conjecture. For every $L\in\mathcal{S}_{\mathrm{poly}}$, one has

Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_\mathbb Q)$, with entire $L$-function … Let…

Let $G$ be an odd-order nilpotent group, with $b_{(H,\phi)}(G)$ and the local factors $\alpha_1,\alpha_2,\alpha_3$ as defined in the surrounding discussion and in Proposition. Expl…

Let $G$ be a nilpotent group, and let $c_{(H,\phi)}(G)$ be the leading constant in the constrained fair-counting asymptotic for $\operatorname{Epi}_{(H,\phi)}(G_{\mathbb{Q}},G)$. O…

Let $K$ be a global field. For all but finitely many primes $\mathfrak p$ of $K$, let $M(\mathfrak p)\in\operatorname{GL}_{r_{\mathfrak p}}(\mathbb C)$ be a unitary matrix, and set…

Let $K$ be a global field and let $\rho$ be an Artin representation with Artin $L$-function $L(s,\rho)$. Set $m=\operatorname{ord}_{s=1/2}L(s,\rho)$. Convergence conjecture. There…

Let $K$ be a global field and let $\pi$ be a nontrivial cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb A)$. For each prime $v$ with Satake parameter $M_\pi(v)$…

Let $K$ be a global field and let $\rho\colon\operatorname{Gal}(K^{\mathrm{sep}}/K)\to\operatorname{GL}_n(\mathbb C)$ be a nontrivial irreducible representation. For a prime $v$ of…

Let $M(p)$ be a unitary matrix of degree $r\in\mathbb N$ for each prime $p$, and let … be an $L$-function that extends to an entire function, satisfies a functional equation with…

Let $\mathcal{S}$ be the Selberg class of functions satisfying the standard Selberg-class axioms, and let $\mathcal{SP}$ be the subclass whose members have a polynomial Euler produ…

Let $X$ be an arithmetic scheme. Its counting function $N(t)$ satisfies $N(1)=\chi_{\mathrm{abs}}(X)$, the absolute Euler characteristic. An absolute Euler product is an infinite p…

Let $\chi\ne\mathbf{1}$ be a Dirichlet character, let $s$ lie on the critical line $\operatorname{Re}s=1/2$, and let $m$ be the order of vanishing of $L(s,\chi)$. Write $\gamma$ fo…

Steuding's conjecture. The value-distribution conclusion above should hold for all $L$-functions with such a polynomial Euler product.

Deep Riemann hypothesis. If $\chi\ne\1$ and $L(s,\chi)\ne0$ with ${\rm Re}(s)=1/2$, then

Euler-product characterization of GRH. If $\chi\ne\1$, then

Let $h(X_1,X_2)\in\mathbb{Z}[X_1,X_2]$ and define … A polynomial $g(U)$ is cyclotomic here if it divides $(1-U^{m})^{n}$ for some positive integers $m$ and $n$. Rudnick–du Sautoy c…

Let $W$ be a type III polynomial: it is as in type II, and there are infinitely many pairs $n,m$ with $c_{n,m}>0$ and $\frac{n+1}{m}>\beta$. These polynomials fall under case (3).…

Let $W(x,y)=\sum_{n,m}a_{n,m}x^ny^m$ be an integral polynomial with $W(x,0)=1$, and define … If $\beta=\max\{\frac{n}{m}:m\geq 1,\ a_{n,m}\neq 0\}$, then the du Sautoy–Woodward con…

Polynomial Euler product conjecture. One has

Let $t$ be a real number tending to infinity, and let the product range over primes $p\leq \log t$. The Riemann zeta function on the line $\operatorname{Re}(s)=1$ is denoted by…