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).
The difference between the harmonic sum and ln(n) approaches γ ≈ 0.5772 as n → ∞. 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.
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…
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) → 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.
Euler-Mascheroni Constant γ is irrational. Its decimal expansion never ends and never repeats. Every digit shown below is computed from the harmonic-logarithm limit.