Problem 4.6

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.

For a nonzero algebraic number $\alpha$, let $f_\alpha(X)=a\prod_{j=1}^d(X-\alpha_j)\in\mathbb Z[X]$ be its primitive irreducible polynomial with positive leading coefficient. Its Mahler measure is $M(\alpha)=a\prod_{j=1}^d\max\{1,|\alpha_j|\}$. For an open $V\subset\mathbb C$, define $L(V)=\inf M(\alpha)^{1/d}$ over nonzero algebraic $\alpha$ which are not roots of unity and whose conjugates all lie outside $V$. Interpret arguments in $(-\pi,\pi]$, and for $0<\theta<\pi$ put $V_\theta=\{z\ne0:|\operatorname{Arg}z|>\theta\}$.

Statement recorded in the supplied source.

Determine the exact value of the Lehmer--Langevin constant $L(V_\theta)$ as a function of every $\theta\in(0,\pi)$. [S2]

Short English statement

Compute the smallest normalized Mahler measure possible when all conjugates avoid the open sector outside the indicated angle.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700051 (AMR-086-0051). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Problem 4.6 and preceding definition of the Lehmer--Langevin constant. https://arxiv.org/abs/math/0312440.