165 problems

Let $\zeta(s)$ denote the Riemann zeta function, and let $\pi$ be the usual circular constant. The odd positive integer values $\zeta(3),\zeta(5),\ldots$ are the values of the zeta…

Chowla–Milnor conjecture. The set $\mathfrak{L}$ is linearly independent over $\mathbb{Q}$. The conjecture is important because its consequences include conditional irrationality r…

Let $n$ be a positive integer and consider the Gamma values … The reflection formula and the Gauss multiplication formula give relations among these values over…

Let $K$ be an algebraic field and consider the Dedekind zeta function of $K$. Generalized Riemann hypothesis. The real part of every non-trivial zero of the Dedekind zeta function…

Let $\mathcal{S}_4$ be the four-star graph, and let $\gamma(\mathcal{S}_4)$ denote its growth parameter. Transcendence conjecture for $\mathcal{S}_4$. The growth parameter…

Grothendieck Period Conjecture. The transcendence degree over $\mathbb{Q}$ of the field generated by its period matrix equals the dimension of the Mumford–Tate group of the Hodge s…

Let $M$ be a 1-motive defined over $\overline{\mathbb{Q}}$, let $\Pi_M$ be its period matrix, and let $\mathcal{G}_{\mathrm{mot}}(M)$ be its motivic Galois group. Grothendieck's pe…

Let $K$ be a CM field that is Galois over $\mathbb{Q}$, and let $\Phi$ be a CM type of $K$. The quantities $p_K(\operatorname{id},\sigma)$ are Shimura period symbols, and…

Folklore conjecture. The numbers $\pi$, $\zeta(3),\zeta(5),\zeta(7),\ldots$ are algebraically independent over the rationals.

Non-algebraicity conjecture. This latter generating series is not algebraic.

Let periods be obtained by integrating algebraic differential forms over algebraic cycles, and consider the formal properties of integration, including additivity, change of variab…

Let $\zeta(s)=\sum_{k=1}^{\infty}k^{-s}$ be the Riemann zeta function, and let $\mathbb{N}$ denote the positive integers. For each $n\in\mathbb{N}$, consider the ratio…

Let $j$ denote the involution of the real line induced by Dyer's outer automorphism of the extended modular group $\operatorname{PGL}(2,\mathbb{Z})$. The map $j$ preserves the set…

Let multiple zeta values be the real numbers … for $d\in\mathbb{N}^{\ast}$ and $s_d,\ldots,s_1\in\mathbb{N}^{\ast}$ with $s_d\geq 2$. The associator equations are the polynomial eq…

Goncharov's direct sum conjecture. The algebra $\mathrm{Z}$ is graded by weights; equivalently, the weight decomposition is a direct sum.

Let $X$ be a smooth projective variety over $\overline{\mathbb Q}$, let $Z_X$ denote the comparison image associated with the periods of $X$, and let $\Omega_X^{\rm mot}$ denote it…

For each integer $p\ge 1$ and each $p$-tuple $\underline{s}=(s_1,\ldots,s_p)$ of integers at least $1$ with $s_1\ge 2$, define the multiple zeta value … The weight is…

Transcendentality–multiplier group conjecture. A quasiperiodic flow $\psi$ is transcendental if and only if

Let $c0$ be the circle constant and let $b6(s)=a3_{n=1}^{\infty}n^{-s}$ denote the Riemann zeta function for integers $s\geq 2$. Algebraic independence conjecture for odd zeta valu…

Let $p$ and $q$ be primes, and let $x$ be a real number. Alaoglu–Erdős conjecture. If both $p^x$ and $q^x$ are rational, then $x$ is an integer. The question was raised in connecti…

Let $\pi$ denote the usual circular constant, and let $x,y,z$ be positive integers. Consider the quantity $|z^\pi-y^\pi-x^\pi|$. Finiteness conjecture for the $\pi$-Fermat expressi…

Minimal-basis conjecture. This set forms a minimal $\mathbf{Q}$-basis for double sums. Such a basis would provide a compact description of the alternating Euler sums appearing in d…

Let $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ be the Riemann zeta function. The values $\zeta(2k+1)$ at odd positive integers are irrational and algebraically independent transcendental…

Let $\mathbf{A}=\mathbb{F}_q[t]$, $A=\mathbb{F}_q[\theta]$, and $K=\operatorname{Frac}(A)=\mathbb{F}_q(\theta)$. Let $E$ be a finite extension of $K$ and let…

A uniformizable abelian $t$-module with everywhere good reduction is an abelian $t$-module satisfying the stated uniformizability and reduction hypotheses. Its Lie algebra contains…