136 problems

Let $\zeta(s)=\sum_{n\ge1}n^{-s}$ be the Riemann zeta function, continued meromorphically to $\mathbb{C}$. Its nontrivial zeros are the zeros $\rho=\beta+i\gamma$ in the critical s…

Let $K/\mathbb{Q}$ be a finite Galois extension, and let $\zeta_K(s)$ be its Dedekind zeta function. Its non-trivial zeros are the zeros other than the trivial zeros arising from t…

Generalized Riemann hypothesis. All non-trivial zeros of $L$-functions lie on the critical line.

Newman's conjecture. All zeros of $\Xi_\aleph(\lambda)$ are real for $\aleph\in\mathbb{R}$. This is the conjecture associated with the de Bruijn–Newman constant, concerning the tra…

Let $\zeta(s)$ be the Riemann zeta function. Its nontrivial zeros are the zeros in the critical strip $0<\operatorname{Re}(s)<1$. Simplicity conjecture. All nontrivial zeros of…

Let $L(f,s)$ be a “nice” $L$-function, and suppose its reciprocal has a Dirichlet-series expansion … valid in some right half-plane. The non-trivial zeros of $L(f,s)$ are the zeros…

Generalized Riemann Hypothesis. The non-trivial zeros of $L(s,f)$ all lie on the critical line

Let $\Lambda_i(V;s)$ be the completed $L$-function associated with a degree-$d_i$, motivic-weight-$i$ $L$-function, including its gamma factors. Generalized Riemann hypothesis. All…

Define the Mertens function by $M(x)=\sum_{n\leq x}\mu(n)$, where $\mu(n)$ is the Möbius function. Weak Mertens conjecture. … The source states that this conjecture implies the Rie…

Let $\mathrm{K}$ be an algebraic number field of degree $n>1$ over $\mathbb{Q}$, let $\mathfrak{o}=\mathfrak{o}_{\mathrm{K}}$ be its ring of integers, and let…

Let … be the Riesz function. Riesz's criterion. The Riemann hypothesis is equivalent to the condition that, for every $\varepsilon>0$ and as $x\to\infty$ through real values, … Thi…

Let the non-trivial zeros of the Riemann zeta function be considered through their pairwise distribution, and let the Gaussian Unitary Ensemble (GUE) be the random-matrix ensemble…

Consider the zeros of the Riemann zeta function and their properly normalized pair correlation function. Montgomery–Odlyzko conjecture. The properly normalized pair correlation fun…

Let the eigenvalue statistics of a hermitian Hamiltonian associated with the Riemann zeros be considered. Montgomery–Odlyzko conjecture. The eigenvalue statistics satisfy random ma…

Let $H(\lambda,z)$ be the De Bruijn–Newman function, and let $\Lambda$ be the De Bruijn–Newman constant, defined so that $H(\lambda,z)$ has only real zeros when…

Let $\zeta(s)$ denote the Riemann zeta function, and let its complex zeros on the critical line be the zeros $s=\frac{1}{2}+it$ of $\zeta(s)$. A self-adjoint operator is an operato…

Let $G$ be a semisimple algebraic group over $\mathbb{Q}$ and $P$ a maximal parabolic subgroup, and let $\xi^{G/P}_{\mathbb{Q}}(s)$ denote the normalized Weng zeta function associa…

Let $\chi$ be a primitive character modulo $q$, and define its Dirichlet $L$-function by … The Extended Riemann Hypothesis. The function $L(s,\chi)$ has no zeros with real part dif…

Let $K={\mathbb Q}(\theta)$ be a number field other than ${\mathbb Q}$, and let $\zeta_K$ denote its Dedekind zeta function, defined from the norms of the ideals of the ring of int…

The Riemann zeta function is defined by analytic continuation from the half-plane where its defining series converges. Its non-trivial zeroes are the zeroes in the strip…

Let $\Xi(\chi,t)=\Lambda(\chi,1/2+it)$ for an abelian character $\chi$ over a number field. The classical absolute-value conjecture. For every $e\geq0$, there are at most two zeroe…

Let $L(x,y)=\prod_i(1-\beta_i^{(y)}/x)$, where the nonzero $\beta_i^{(y)}$ are the zeroes of $L(x,y)$, and let $\lambda_i^{(y)}:=1/\beta_i^{(y)}$ be the reciprocal zeroes. The char…

Let $L(s)$, with $s=(x,y)\in S_\infty$, be a characteristic-$p$ $L$-series arising from arithmetic, and let $\mathbf K_L$ be the finite extension of $\mathbf K_{\mathbf V}$ generat…

Let $\mathfrak L=\mathbb Q$ and let $\chi$ be an abelian character. Let $\Lambda(\chi,s)$ be the completed $L$-series, with its functional equation relating $\Lambda(\chi,s)$ and…

The Generalized Riemann Hypothesis. The zeroes of $\Lambda(\chi,s)$ lie on the line $\{s=1/2+it\mid t\in\mathbb R\}$, equivalently, the zeroes of $\Xi(\chi,t)$ are real.