34 problems

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 $Z_F$ be the fractal zero set constructed from the positive imaginary parts of the non-trivial zeros of $\zeta(s)$ by the recursive four-subinterval construction described in t…

Let $(\mathcal{D}_{\mathrm{arith}},\{\eta_p\})$ be a chiral adelic Dirac system, where the coefficient functions $\eta_p$ are even, and let $\xi_{\mathrm{Dirac}}$ denote its spectr…

Let $(C,0)$ be a planar curve singularity, and let $L(q,t,0)$ be the specialization at $a=0$ of its Hilbert $L$-function. A $t$-zero is a complex number $\xi$ such that…

Spira's conjecture. All the non-trivial zeros of $Z_{N(t)}(t)$ are real.

Let $p$ and $q$ be primes with $p\equiv q\equiv 3 \pmod 8$, satisfying $0<2a<p<q$, and let $\chi_q(p)=\left(\frac{p}{q}\right)$ be the Legendre symbol modulo $q$. Let … holds for a…

Let $q$ be a prime with $q\equiv 3 \pmod 8$, let $x$ be rational with $0<x<1/2$, and define … where $h(q)$ is the class number and $\chi_q$ is the relevant quadratic character. Num…

Let $g_n$ be an isolated Gram point, and let $\gamma^{\mathrm{descend}}$ denote the descending curve in the two-dimensional deformation associated with the shifting and descending…

For $N\in\mathbb{N}$, let $\mathcal{Z}_N$ be the parameter space of functions … with $\overline{a}\in\mathbb{R}^N$. A path $\gamma(r)$ in this space starts at the core function…

Let $\alpha\in\mathbb{N}$, let $\zeta_Q(s)$ be the Epstein zeta function associated with a real-valued positive definite $\alpha\times\alpha$ matrix $Q$, and let…

Let $\lambda_j=\eta_j^2$ be the transformed nontrivial zeta zeros, ordered as in the source. For a parameter $\gamma$ with … consider consecutive transformed zeros that are not con…

Let $\rho_k$ be the nontrivial zeros of the Riemann zeta function with positive imaginary part, and set … Let $\Lambda$ be the associated entire function, and let its spacing const…

First Hankel relaxation Riemann hypothesis.

$N$-th Hankel relaxation Riemann hypothesis. The function $\Lambda$ is in the $N$-th order Laguerre–Pólya class. These are proposed relaxations of the Riemann hypothesis: membershi…

Let $\lambda_n$ denote the Li coefficients of the Riemann zeta function. In view of the recurrence relation for these coefficients, strict recurrence inequality for Li coefficients…

Let $S$ be the Selberg class, let $\mathcal L\in \widetilde{\mathcal S}\cap S$, let $K$ be a compact subset of the strip $\mathfrak S=\{\sigma_m<\sigma<1\}$ with connected compleme…

Semi-local framework conjecture. The semi-local operator theoretic framework with $S:=\{\infty\}\cup \{p\mid p<q\}$ suffices to prove the Weil inequality for all test functions wit…

Let $\operatorname{RM}$ be the random model, with a realisation determined by the shifts $a$ and factors $\rho_k(n+a)$, and let $\pi_{\operatorname{RM}}(x)$ be its counting functio…

Let $N$ be a positive integer, let $\rho_1$ and $\rho_2$ range over the nontrivial zeros of the Riemann zeta function, and let $B_x(a,b)$ denote the incomplete beta function. The n…

Weak Generalized Riemann Hypothesis. There exists a constant $\frac{1}{2}>\epsilon\geq 0$ such that all nontrivial roots of $L(s,\chi)$ lie in

Soundararajan's conjecture. The following two statements are equivalent:

Let $\Xi_t(x)$ be the deformed Riemann zeta function, and let $\Lambda\in\mathbb R$ be the De Bruijn–Newman constant, characterized by the fact that $\Xi_t$ has only real zeros for…

Functional-equation conjecture. For every $s\in\mathbb{C}$ with $0<\operatorname{Re}(s)<1$,