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Franz Mertens (March 20, 1840 – March 5, 1927) was a German mathematician. He was born in Środa in the Grand Duchy of Poznań, Kingdom of Prussia (now Środa Wielkopolska, Poland) and died in Vienna, Austria.

Franz Mertens

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 x1/2, 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.