20 March 1840|
|Died||5 March 1927
|Institutions||University of Vienna
|Alma mater||University of Berlin|
|Doctoral advisor||Ernst Eduard Kummer
|Doctoral students||Ernst S. Fischer
|Known for||Mertens function
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.