Algebraic independence conjecture for odd zeta values

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The algebraic independence conjecture for odd zeta values

   The constants involved are $c0$ and the odd zeta values $b6(2n+1)$ for $ngeq 1$.

   Algebraic independence conjecture. The numbers $c0$ and $b6(2n+1)$, for $ngeq 1$, are algebraically independent over $\mathbb{Q}$.

   This conjecture expresses the expectation that odd zeta values do not satisfy algebraic relations with $c0$ or with one another; unlike the even zeta values, their arithmetic nature remains largely mysterious.

   source: Clément Dupont, “Valeurs zêta multiples”, arXiv:2109.01699 (2021).

2. The algebraic independence conjecture for odd zeta values

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

   $$\pi,\zeta(3),\zeta(5),\dots$$

   are all algebraically independent over $\mathbb Q$. This is a folklore conjecture motivated by the absence of known transcendence results for odd zeta values beyond irrationality results such as Apéry's theorem for $\zeta(3)$ and the theorem of Ball and Rivoal that infinitely many odd zeta values are irrational.

   source: Bo-Hae Im, Hojin Kim, Khac Nhuan Le, Tuan Ngo Dac and Lan Huong Pham, “Hopf algebras and multiple zeta values in positive characteristic”, arXiv:2301.05906 (2023).