Meissel–Mertens constant
The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as the Mertens constant, Kronecker's constant (after Leopold Kronecker), Hadamard–de la Vallée-Poussin constant (after Jacques Hadamard and Charles Jean de la Vallée-Poussin), or the prime reciprocal constant, is a mathematical constant in number theory, defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm: M = lim n → ∞ ( ∑ p prime p ≤ n 1 p − ln ( ln n ) ) = γ + ∑ p [ ln ( 1 − 1 p ) + 1 p ] . {\displaystyle M=\lim _{n\rightarrow \infty }\left(\sum _{\scriptstyle p{\text{ prime}} \atop \scriptstyle p\leq n}{\frac {1}{p}}-\ln(\ln n)\right)=\gamma +\sum _{p}\left[\ln \! \left(1-{\frac {1}{p}}\right)+{\frac {1}{p}}\right].} Here γ is the Euler–Mascheroni constant, which has an analogous definition involving a sum over all integers (not just the primes).