Euler-Mascheroni Constant Irrationality

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 $H_N=\sum_{j=1}^N1/j$ and let $\log$ denote the natural logarithm. The Euler--Mascheroni constant is the real limit \[ \gamma=\lim_{N\to\infty}(H_N-\log N). \] A real number is irrational if it is not in $\mathbb Q$. It is transcendental if it is not a root of any nonzero polynomial in $\mathbb Z[X]$; transcendence implies irrationality.

Statement recorded in the supplied source.

Determine both of the following assertions: \[ \gamma\notin\mathbb Q; \qquad P(\gamma)\ne0\quad\text{for every nonzero }P\in\mathbb Z[X]. \] Thus, is $\gamma$ irrational, and more strongly, is it transcendental? [S2] [S3]

Short English statement

Is the Euler--Mascheroni constant irrational? Is it also transcendental?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1511 (NUM-017). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jonathan Sondow, Criteria for Irrationality of Euler's Constant, Proceedings of the American Mathematical Society 131 (2003), 3335--3344; arXiv:math/0209070v2, Section 1, equation (1). https://arxiv.org/abs/math/0209070.
• S3 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (2004), Section 2.3, the Euler-constant irrationality question, and Section 3.1, the stronger non-period/transcendence question. https://arxiv.org/abs/math/0312440.