30 problems

Let $y1, ldots,yn$ be solutions of a fixed Painlevé equation whose complex parameters are generic. Write $operatorname{td}(L/K)$ for the transcendence degree of a field extension…

For a $d$-tuple of positive integers $\underline{n}=(n_1,\dots,n_d)$ with $n_1\geq 2$, define the multiple zeta value … and define its weight by…

Let $K$ be the base field, let $\Omega$ be the domain of the Drinfeld modular forms, and let $t(z)$ be the usual parameter at infinity for $z\in\Omega$. The functions $E$, $g$, and…

Three-logarithm conjecture. One has

For positive integers $s\ge 2$, define the Riemann zeta function by … The values at positive even integers are known to be rational multiples of powers of $\pi$, so the conjecture…

The four numbers … are considered, where $\Gamma$ is Euler's Gamma function. Algebraic-independence conjecture. At least three of these four numbers are algebraically independent.…

Let $X\subseteq\mathbb R^n$ be definable in a sharply o-minimal structure with sharp derivatives, and let $k=0,\ldots,n-\dim X-1$. For $g,h\in\mathbb N$, let…

Let $X$ be a set definable in a sharply o-minimal structure, and let $V$ range over algebraic varieties over $mathbb Q$ of dimension $k<\operatorname{codim} X$. Set…

Let $N_\tau$ denote the number of occurrences of a binary pattern (word) $\tau$ in a sequence. For sequences of length up to $k$, let $A(k)$ be the number of algebraically independ…

Let $T_p(z)$ denote the valuation generating function associated with a prime $p$. Let $p_1,p_2,\dots,p_m$ be distinct primes, and let $\alpha_1,\alpha_2,\dots,\alpha_m$ be non-zer…

Let $\alpha \in \overline{\mathbb{Q}}$ with $\alpha \not=0,1$, and let $x_1,\dots,x_n \in \overline{\mathbb{Q}}$ be such that $1,x_1,\dots,x_n$ are $\mathbb{Q}$-linearly independen…

Algorithmic simplification conjecture. If one has prior knowledge of the degree of $f^*$ or $h^*$, the repeated-training and averaging step in Algorithm can be removed, because gen…

Let $\overline{\mathcal{Z}}$ be the $\overline{k}$-algebra generated by all $\infty$-adic MZVs, and let $\overline{\mathcal{Z}}_v$ be the corresponding $v$-adic algebra. The natura…

The numbers $\pi,\zeta(3),\zeta(5),\dots$ are classical zeta values, with $\pi$ the usual circular constant. Algebraic independence conjecture. The numbers … are all algebraically…

Let $r\geq 1$ be an integer. Let $b_1,\ldots,b_r$ be multiplicatively independent positive integers, and, for every $i$, $1\leq i\leq r$, let $\xi_i$ be a real number that is autom…

Let $\overline{\mathfrak{Z}}$ be the $\overline{\mathbb{Q}}$-algebra generated by multizeta values, and let $\overline{\mathfrak{Z}}_w$ be the $\overline{\mathbb{Q}}$-vector space…

Let $r\geq 2$, let $\rho_1,\ldots,\rho_r$ be pairwise multiplicatively independent algebraic numbers, and, for each $i$ with $1\leq i\leq r$, let $\mathbf a_i$ be a $\rho_i$-morphi…

Let $r\geq 2$ be an integer. For each $i$ with $1\leq i\leq r$, let $q_i\geq 2$ be an integer, let $f_i(z)\in \overline{\mathbb Q}\\{z\\}$ be a $q_i$-Mahler function, and let…

Let $\alpha_1,\ldots,\alpha_n$ be non-zero algebraic numbers such that the numbers $\log \alpha_1,\ldots,\log \alpha_n$ are $\mathbb{Q}$-linearly independent. Weak Schanuel conject…

For $k\ge0$, let … be the $\calA_1$-Bernoulli numbers, and for $k\ge2$ let … be the $k$th $\calA_1$-Fermat quotient. Set $\gb_1=1$. Suppose $n_1,\dots,n_r\in\N$ and…

Let $\zeta_{\mathbb{Z}}(n)=\sum_{m\geq 1}m^{-n}$ denote the Riemann zeta value for an integer $n\geq 2$, and let $\overline{\mathbb{Q}}$ be the field of algebraic numbers. Odd zeta…

Schanuel Subset Conjecture. If the set $\{\alpha_1,\dotsc,\alpha_n,e^{\alpha_1},\dotsc,e^{\alpha_n}\}$ is $\overline{\mathbb{Q}}$-dependent on a subset $\{\beta_1,\dotsc,\beta_n\}$…

Gelfond's Power Tower Conjecture. The number ${}^kz$ is transcendental for every $k\geqslant2$. Moreover, when $z:=e^{\omega}$, the numbers ${}^1z,{}^2z,{}^3z,\dots$ are algebraica…

Let $\mathcal{L}$ be the $\mathbb Q$-vector space of logarithms of algebraic numbers: … For $n\geq 2$, let ${\sf LD}_n^{\mathbb Q}$ denote the relation of $n$ elements of…

Let $\Lambda\subset\mathbb{C}$ be a lattice, let $\lambda\in\Lambda\setminus\{0\}$, and let $\omega\in\mathbb{C}\setminus(\mathbb{Q}\lambda\cup\Lambda)$. Waldschmidt's conjecture.…