What the constant is

Euler's constant $\gamma$ measures the gap between the harmonic numbers and the logarithm:

$$\gamma \;=\; \lim_{n\to\infty}\left(\sum_{k=1}^{n}\frac{1}{k} \;-\; \log n\right) \;=\; 0.5772156649\ldots$$

It appears wherever the harmonic series is compared against an integral, and it is the value at $1$ of the digamma function, $\psi(1) = -\gamma$, equivalently $\gamma = -\Gamma'(1)$. Among the classical constants of analysis it is the conspicuous one whose arithmetic nature is unknown.

What is being asked

For $\pi$ and $e$ the arithmetic questions were settled long ago: both are irrational and transcendental. For $\gamma$, neither is known. It is not known whether $\gamma$ is irrational, and a fortiori not whether it is transcendental, though it is universally expected to be both.

The goal theorem of this mission is transcendence,

$$\gamma \notin \overline{\mathbb{Q}},$$

with irrationality carried as a separate, weaker target — a proof of transcendence yields irrationality immediately, but not conversely, and irrationality alone would already be a landmark.

What is actually known

Progress has come in three forms, and the milestones below formalize each.

Conditional bounds on a putative denominator. If $\gamma$ were rational, its denominator would have to be enormous. Brent and McMillan (1980), computing $\gamma$ to $30{,}000$ places by an algorithm built on modified Bessel functions, showed any denominator exceeds $10^{15000}$; a continued-fraction analysis by Papanikolaou (1997) pushed this past $10^{244663}$. These are not steps toward a proof so much as a measurement of how far brute computation can go.

Disjunctive results. The strongest unconditional statements pair $\gamma$ with the Euler–Gompertz constant

$$\delta \;=\; \int_0^{\infty} \frac{e^{-u}}{1+u}\, du \;=\; 0.5963473623\ldots$$

Aptekarev, building on work of Mahler and Shidlovskii, observed that at least one of $\gamma$ and $\delta$ is irrational. Rivoal later strengthened this to at least one of them is transcendental. Neither argument isolates which, and that is precisely the obstruction: the Padé-approximation machinery that controls the pair does not separate them.

Irrationality criteria. Sondow, adapting Beukers' treatment of Apéry's theorem for $\zeta(3)$, gave criteria equivalent to the irrationality of $\gamma$ in terms of the fractional parts of certain integer sequences. They reformulate the problem rather than resolve it.

Timeline

• 1734 — Euler introduces the constant and computes it to six decimals.
• 1790s–1800s — Mascheroni computes further digits; the constant acquires its second name.
• 1873 — Hermite proves $e$ transcendental; 1882 — Lindemann does the same for $\pi$. The methods do not reach $\gamma$.
• 1980 — Brent and McMillan: if $\gamma = p/q$ then $q > 10^{15000}$.
• 1997 — Papanikolaou: the same denominator exceeds $10^{244663}$.
• 2009 — Aptekarev: at least one of $\gamma$, $\delta$ is irrational.
• 2012 — Rivoal: at least one of $\gamma$, $\delta$ is transcendental.
• 2010s — Murty, Saradha and others obtain transcendence results for generalized Euler–Lehmer constants, again leaving $\gamma$ itself untouched.

Formalization notes

Mathlib provides the constant as Real.eulerMascheroniConstant, defined as the limit of $\sum_{k\le n} 1/k - \log n$, together with the identifications $\psi(1) = -\gamma$ and $\gamma = -\Gamma'(1)$ and the numeric bounds $1/2 < \gamma < 2/3$. Irrational and Transcendental ℚ are Mathlib's standard predicates. The Euler–Gompertz constant is not in Mathlib and is supplied here as a mission definition.