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).
The gap between the prime reciprocal sum and ln(ln(n)) converges to M ≈ 0.26149. Both sums diverge, but their difference settles to a constant, just as 1+1/2+⋯+1/n minus ln(n) settles to γ.
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.
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…
The harmonic series (blue) grows like ln(n). The prime reciprocal sum (red) grows like ln(ln(n)), an extraordinarily slow doubly-logarithmic growth. At n=10^100, ln(ln(10^100)) ≈ ln(230) ≈ 5.4. Prime gaps are captured in the constants γ and M respectively.
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.