Who discovered the Euler-Mascheroni constant?

Leonhard Euler first described it in 1735 while studying the difference between the harmonic series and the natural logarithm. Lorenzo Mascheroni computed it to more decimal places in 1790 and gave it the symbol \u03B3.

Is \u03B3 rational or irrational?

Nobody knows. This is one of the most famous open problems in mathematics. Mathematicians strongly suspect \u03B3 is irrational, but no proof has ever been found despite centuries of effort. It has been computed to over 600 billion decimal places.

What is the harmonic series?

The harmonic series is 1 + 1/2 + 1/3 + 1/4 + ... This sum grows without bound - proven by Nicole d'Oresme around 1350 AD - but grows very slowly. \u03B3 measures the precise gap between this sum and ln n as n grows toward infinity.

Why is \u03B3 considered mysterious compared to \u03C0 and e?

\u03C0 and e have known geometric meanings and are both proven transcendental. \u03B3 arises from a subtler comparison between a discrete sum and a continuous function. Even its most basic property - whether it can be a fraction - remains unsolved.

Where does \u03B3 appear in science?

\u03B3 appears in the distribution of prime numbers, in the Riemann zeta function near s=1, in the Gamma function at integer values, in Bessel functions important to wave physics, and in antenna signal coverage calculations.

Christoffer De Geer March 20, 2026

What is the Euler-Mascheroni Constant (γ)?

γ = lim (1 + 1/2 + ⋯ + 1/n) - ln(n)

γ ≈ 0.57721566490153286060. Computed to 600 billion digits. Irrationality unknown.

The harmonic series 1 + 1/2 + 1/3 + 1/4 + ⋯ diverges, but it grows incredibly slowly. After a million terms it barely reaches 14. The natural logarithm ln(n) grows at the same rate. The Euler-Mascheroni constant γ is the precise gap between them: γ = lim (1 + 1/2 + 1/3 + ⋯ + 1/n) - ln(n).

H(n) − ln(n) converges to the Euler-Mascheroni constant γ

The difference between the harmonic sum and ln(n) approaches γ ≈ 0.5772 as n to ∞. Convergence is very slow - the gap is still 0.001 at n = 1000.

γ appears throughout analysis and number theory. It links the harmonic series to the Riemann zeta function: γ = -ζ'(1) in a formal sense. It appears in the Gamma function Γ'(1) = -γ, in the distribution of prime gaps, in Bessel functions, and in the asymptotic expansion of the digamma function.

Key facts about γ

γ = lim(n to ∞) [H(n) − ln(n)] ≈ 0.5772156649…

γ = −Γ'(1) = −∫₀^∞ e⁻ˣ ln(x) dx

Whether γ is irrational is unknown - one of the oldest open problems in mathematics.

Whether γ is rational or irrational is one of the oldest open problems in mathematics. Almost every mathematician believes it is transcendental, but no proof exists. It has been computed to over 600 billion decimal places: 0.57721566490153286060651209008240243…

Harmonic Series Meissel-Mertens Constant Riemann Zeta Function Stirling's Approximation

Harmonic staircase H(n) versus smooth ln(n) + γ

The harmonic partial sums H(n) (red, stepped) versus ln(n)+γ (blue, smooth). The gap between them approaches 0 but oscillates: H(n)−ln(n) to γ.

Key facts about Euler-Mascheroni Constant γ

The Euler-Mascheroni constant gamma is approximately 0.57721566490153286060. Whether it is rational or irrational is unknown, one of the most famous open problems in mathematics. Euler first published it in 1734; Mascheroni computed it independently in 1790. Gamma appears in the Gamma function, the Riemann zeta function, Mertens theorem on prime products, Bessel functions, and the distribution of prime gaps. Since no streaming algorithm exists, its digits are pre-computed and stored.

FAQ

Who discovered the Euler-Mascheroni constant?

Leonhard Euler first described it in 1735 while studying the difference between the harmonic series and the natural logarithm. Lorenzo Mascheroni computed it to more decimal places in 1790 and gave it the symbol \u03B3.
Is \u03B3 rational or irrational?

Nobody knows. This is one of the most famous open problems in mathematics. Mathematicians strongly suspect \u03B3 is irrational, but no proof has ever been found despite centuries of effort. It has been computed to over 600 billion decimal places.
What is the harmonic series?

The harmonic series is 1 + 1/2 + 1/3 + 1/4 + ... This sum grows without bound - proven by Nicole d'Oresme around 1350 AD - but grows very slowly. \u03B3 measures the precise gap between this sum and ln n as n grows toward infinity.
Why is \u03B3 considered mysterious compared to \u03C0 and e?

\u03C0 and e have known geometric meanings and are both proven transcendental. \u03B3 arises from a subtler comparison between a discrete sum and a continuous function. Even its most basic property - whether it can be a fraction - remains unsolved.
Where does \u03B3 appear in science?

\u03B3 appears in the distribution of prime numbers, in the Riemann zeta function near s=1, in the Gamma function at integer values, in Bessel functions important to wave physics, and in antenna signal coverage calculations.

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