Conjecture 3.19 — Rohrlich

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G(z)=\Gamma(z)/\sqrt{2\pi}$ and let $H=\mathbb C^\times/\overline{\mathbb Q}^{\times}$, regarded as an abelian group. The functional equation of $\Gamma$ makes $g(x)=[G(x)]$ a well-defined map on $A=(\mathbb Q/\mathbb Z)\setminus\{0\}$. An odd distribution into an additive abelian group $B$ is a map $f:A\to B$ satisfying $f(-x)=-f(x)$ and $\sum_{j=0}^{N-1}f(x+j/N)=f(Nx)$ whenever $N\ge1$ and $Nx\ne0$ in $\mathbb Q/\mathbb Z$. Let $U$ be the free $\mathbb Z[1/2]$-module on symbols $[x]$ for $x\in A$, modulo exactly these oddness and distribution relations.

Statement recorded in the supplied source.

Is the natural map $U\to H$, sending $[x]$ to $g(x)$, an isomorphism onto the subgroup generated by the $g(x)$ over $\mathbb Z[1/2]$? Equivalently, does every odd distribution with values in a group in which multiplication by $2$ is bijective factor uniquely through this gamma-value subgroup? [S2]

Short English statement

Are the gamma functional equations, reflection and multiplication relations the complete multiplicative relations among normalized gamma values at rational nonintegral points, after allowing division by two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700037 (AMR-086-0037). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 3.19 and its defining distribution equations. https://arxiv.org/abs/math/0312440.