Euler-Mascheroni Constant γ Calculator

The harmonic sum Hₙ grows like ln n, and the leftover gap converges to γ ≈ 0.5772156649015329. At n = 100: H₁₀₀ = 5.187378 while ln 100 = 4.605170, difference 0.582208 — already within 0.005 of γ, with the error decaying like 1/(2n). The calculator computes Hₙ and ln n for your n, shows their difference approaching 0.5772156649, and accelerates convergence with the expansion Hₙ ≈ ln n + γ + 1/(2n) − 1/(12n²), reaching 10-digit accuracy by n ≈ 10⁶. Unlike π or e, γ’s status is open: nobody knows whether it is irrational. The constant threads through number theory and analysis — Euler’s gamma function satisfies Γ′(1) = −γ, and the exponential integral carries ln x + γ. Known bounds (0.57 < γ < 0.58) and record digit counts (billions) are noted for context.

1. 1

   Enter n

   The harmonic cutoff

2. 2

   Compute Hₙ

   1 + ½ + … + 1/n summed (H₁₀₀ = 5.187378)

3. 3

   Subtract ln n

   The difference approaches γ

4. 4

   Apply acceleration

   The 1/(2n) − 1/(12n²) refinement tightens the estimate

• 1

  γ ≈ 0.5772156649 sits between Hₙ and ln n + 1

• 2

  Gap decays like 1/(2n): doubling n halves the error

• 3

  Irrationality of γ remains unproven

• Confusing γ with e ≈ 2.71828 or Γ(γ) notations

• Expecting Hₙ − ln n to reach γ exactly at some n

• Ignoring the slow 1/(2n) convergence and trusting few-digit estimates

• Mixing γ with the golden ratio φ ≈ 1.618