h1card_01p_02
Prime reciprocal sum grows like ln(ln(n)) + M
Σ_{p≤n} 1/p ≈ ln(ln(n)) + M
M ≈ 0.2615 (Meissel-Mertens constant)
At n=10: ≈ 0.84 n=100: ≈ 1.18 n=1000: ≈ 1.52 n=10^10: ≈ 2.30
Compared to harmonic sum Σ 1/n ≈ ln(n) + γ — prime reciprocals grow far slower.
p_04
M vs γ: two gap constants
Side by side comparison of Euler-Mascheroni and Meissel-Mertens constants
| Euler-Mascheroni γ | Meissel-Mertens M |
|---|
| Σ 1/n − ln(n) → 0,5772 | Σ 1/p − ln(ln n) → 0,2615 |
| alle ganzen Zahlen | nur Primzahlen |
p_06links
Harmonic sum vs prime reciprocal sum: both diverge, at very different rates
Harmonic sum (blue): 2.93, 5.19, 7.49, 9.79. Prime reciprocal sum (grows like ln(ln(n))+M): only 0.84, 1.18, 1.52, 1.85 at the same points.
card_08card_09
Question
Berapa nilai numerik M?
tap · space
1 / 10