Sum the reciprocals of all primes up to n: 1/2 + 1/3 + 1/5 + 1/7 + ⋯ + 1/p. This grows, but extraordinarily slowly: as ln(ln(n)). The Meissel-Mertens constant M is the precise gap between this sum and its dominant term, just as the Euler-Mascheroni constant γ is the gap between the harmonic series and ln(n).
Euler proved in 1737 that the sum of all prime reciprocals diverges. This is much harder than proving there are infinitely many primes, and gives a quantitative sense of how dense primes are. Mertens's theorem then says Σ(p≤n) 1/p = ln(ln(n)) + M + O(1/log n), giving M as the precise constant term.
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 |
| All integers | Primes only |
M and γ are related by M = γ + Σₚ(ln(1−1/p) + 1/p). Whether either constant is irrational is unknown. They are both computed to billions of decimal places and believed transcendental, but no proof exists for either. M: 0.261497212847642783755426838608669…
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.
The Euler-Mascheroni constant gamma measures the gap between the harmonic series (1 + 1/2 + 1/3 + ... + 1/n) and ln(n). The Meissel-Mertens constant M plays the same role for the sum of prime reciprocals (1/2 + 1/3 + 1/5 + ... + 1/p) versus ln(ln(n)). Both are the "error correction" constants for divergent series that grow logarithmically.
The Meissel-Mertens constant M ≈ 0.26149 plays the same role for prime reciprocals as the Euler-Mascheroni constant plays for the harmonic series. Mertens proved in 1874 that 1/2 + 1/3 + 1/5 + ... + 1/p = ln(ln(n)) + M + small error. Whether M is irrational is unknown. It appears in Mertens' theorem on prime products and in the density of smooth numbers. M and gamma are related by a specific sum over all primes.