Franz Mertens was a German

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

. He was born in Środa

Sroda Wielkopolska

Środa Wielkopolska is a town in central Poland, about southeast of Poznań, with 22,001 inhabitants .-Overview:It is situated in Greater Poland Voivodeship, having previously been in the former Poznań Voivodeship...

 in the Grand Duchy of Poznań

Grand Duchy of Poznan

The Grand Duchy of Posen, or the Grand Duchy of Poznań was part of the Kingdom of Prussia, created from territories annexed by Prussia after the Partitions of Poland, and formally established following the Napoleonic Wars in 1815. Per agreements derived at the Congress of Vienna it was to have...

, Kingdom of Prussia

Kingdom of Prussia

The Kingdom of Prussia was a German kingdom from 1701 to 1918. Until the defeat of Germany in World War I, it comprised almost two-thirds of the area of the German Empire...

 (now Środa Wielkopolska, Poland

Poland

Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...

) and died in Vienna

Vienna

Vienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...

, Austria

Austria

Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...

.

The Mertens function M(x) is the sum function for the Möbius function

Möbius function

The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...

, in the theory of arithmetic function

Arithmetic function

In number theory, an arithmetic function is a real or complex valued function ƒ defined on the set of natural numbers In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers In number theory, an arithmetic (or...

s. The Mertens conjecture

Mertens conjecture

In mathematics, the Mertens conjecture is the incorrect statement that the Mertens function M is bounded by √n, which implies the Riemann hypothesis...

 concerning its growth, conjecturing it bounded by x^1/2, which would have implied the Riemann hypothesis

Riemann hypothesis

In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

, is now known to be false (Odlyzko

Andrew Odlyzko

Andrew Michael Odlyzko is a mathematician and a former head of the University of Minnesota's Digital Technology Center.In the field of mathematics he has published extensively on analytic number theory, computational number theory, cryptography, algorithms and computational complexity,...

 and te Riele, 1985). The Meissel–Mertens constant is analogous to the Euler–Mascheroni constant

Euler–Mascheroni constant

The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

, but the harmonic series

Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...

 sum in its definition is only over the primes

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 rather than over all integers and the logarithm is taken twice, not just once.

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