Euler-Mascheroni constant

Definitions and mathematical statement

The Euler--Mascheroni constant is the real number \[ \gamma=\lim_{n\to\infty}\left(\sum_{j=1}^{n}\frac1j-\log n\right), \] where $\log$ is the natural logarithm. The limit fixes a particular constant; it is not Euler's number $e$ or a value of the gamma function. A real number is algebraic over $\mathbb Q$ if it is a zero of a nonzero polynomial with integer coefficients, and is transcendental otherwise. The source asks whether $\gamma$ is transcendental: \[ \forall d\ge0\ \forall(a_0,\ldots,a_d)\in\mathbb Z^{d+1} \setminus\{(0,\ldots,0)\},\qquad \sum_{j=0}^{d}a_j\gamma^j\ne0. \] Clearing denominators gives the same condition for polynomials with rational coefficients. A positive answer must exclude polynomials of every degree, with no bound on their coefficients. Irrationality would exclude only nonzero linear polynomials and would be a weaker conclusion. A negative answer must produce a nonzero polynomial annihilating $\gamma$ with a rigorous identity, rather than a numerical near-equality. This is an exact arithmetic question; any finite decimal approximation alone cannot distinguish an algebraic constant from a transcendental one.

Short English statement

Is the Euler--Mascheroni constant the root of no nonzero polynomial with integer coefficients?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3370 (OPG-37329), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Euler-Mascheroni constant. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/euler_mascheroni_constant.