Page "Franz Mertens" Paragraph 1

The Mertens function M ( x ) is the sum function for the Möbius function, in the theory of arithmetic functions.

The Mertens conjecture concerning its growth, conjecturing it bounded by x < sup > 1 / 2 </ sup >, which would have implied the Riemann hypothesis, is now known to be false ( Odlyzko and te Riele, 1985 ).

The Meissel – Mertens constant is analogous to the Euler – Mascheroni constant, but the harmonic series sum in its definition is only over the primes rather than over all integers and the logarithm is taken twice, not just once.