14 problems

Generalized Riemann Hypothesis. All zeros of Dirichlet $L$-functions lie on the critical line

Let $L(s,\pi)$ be a meromorphic $L$-function satisfying the stated Dirichlet-series, Euler-product, analytic-continuation, and functional-equation properties, and let … be its comp…

Let $L(s,f)$ be an automorphic $L$-function, and let the corresponding modified renormalization-group flow have fixed point $(I_P^{(f)*},I_Z^{(f)*})$ with asymmetry ratio … Here…

Fujii's conjecture. For every $\alpha>0$, replacing $e^{-p/x}$ by $e^{-(p/x)^\alpha}$ gives a statement equivalent to the Generalized Riemann Hypothesis for $L(s,\chi_4)$; likewise…

Artin GRH. Artin $L$-functions should satisfy the generalized Riemann hypothesis and therefore have no zeros with real part greater than $1/2$.

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 $A$ be an abelian variety over a number field $k$, let $\operatorname{ST}(A)$ be its Sato–Tate group, and let $\chi$ be an irreducible character of $\operatorname{ST}(A)$. Let…

Let $k$, $q$, and $N$ be positive integers with $N$ squarefree, $k$ even, and $\gcd(N,q)=1$. Let $f\in S_k^{\mathrm{new}}(\Gamma_0(N))$ be a non-CM newform, and let $\chi$ be a pri…

Let $L(s,\chi)$ be a Dirichlet $L$-function, and consider its nontrivial zeros. Generalized Riemann hypothesis. Every nontrivial zero of every Dirichlet $L$-function lies on the cr…

Symmetric-power conjecture. For every $k\geq0$, the conductor of $L(s,\operatorname{Sym}^k E)$ is $N_E^k$; its gamma factor is

Let $f\in S_k^{\operatorname{new}}(\Gamma_0(N))$ be a newform with squarefree level $N$. For each nonnegative integer $n$, let $L(\operatorname{Sym}^n f,s)$ be the symmetric-power…

Let $\varpi:\mathbb{N}\to\{0,1\}$ be the characteristic function of odd prime numbers, and consider the weight function … Fujii established the corresponding generalization of Cheb…

Let $\kappa_4$ be the nontrivial real Dirichlet character modulo $4$, with Dirichlet $L$-function … Let $\delta(x,4)=\pi(x;4,3)-\pi(x;4,1)$ denote Chebyshev's bias, and let…

Let $K$) be a Galois cubic number field with discriminant $\Delta$. The field $K$ is norm-Euclidean if for every $\alpha,\beta\in\mathcal{O}_K$ with $\beta\neq 0$, there exists…