Franz Mertens

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Franz Mertens
Franz Mertens.jpg
Born (1840-03-20)20 March 1840
Schroda, Prussia
Died 5 March 1927(1927-03-05) (aged 86)
Vienna, Austria
Nationality Germany
Fields Mathematics
Institutions University of Vienna
Alma mater University of Berlin
Doctoral advisor Ernst Eduard Kummer
Leopold Kronecker
Doctoral students Ernst S. Fischer
Eduard Helly
Tonio Rella
Known for Mertens function
Meissel–Mertens constant
Mertens' theorems

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

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. Mertens' theorems are three 1874 results related to the density of prime numbers.

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