Franz Mertens (20 March 1840 – 5 March 1927) (also known as Franciszek Mertens) was a Polish mathematician. He was born in Schroda in the

Grand Duchy of Posen The Grand Duchy of Posen (german: Großherzogtum Posen; pl, Wielkie Księstwo Poznańskie) was part of the Kingdom of Prussia, created from territories annexed by Prussia after the Partitions of Poland, and formally established following the ...

, Kingdom of Prussia (now

Środa Wielkopolska Środa Wielkopolska (until 1968 ''Środa''; german: Schroda) is a town in western-central Poland, situated in the Greater Poland Voivodeship, about southeast of Poznań, with 22,001 inhabitants (2009). It is the seat of Środa Wielkopolska County ...

, Poland) and died in Vienna,

Austria Austria, , bar, Östareich officially the Republic of Austria, is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine states, one of which is the capital, Vienna, the most populous c ...

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

arithmetic function In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their ...

s. The

Mertens conjecture In mathematics, the Mertens conjecture is the statement that the Mertens function M(n) is bounded by \pm\sqrt. Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1 ...

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

Meissel–Mertens constant The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamard– de la Vallée-Poussin constant or the prime reciprocal constant, is a mathematical constant in n ...

is analogous to the Euler–Mascheroni constant, but the harmonic series sum in its definition is only over the

primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

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

Mertens's theorems In number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.F. Mertens. J. reine angew. Math. 78 (1874), 46–6Ein Beitrag zur analytischen Zahlentheorie/ref> "Mertens' theorem" may al ...

are three 1874 results related to the density of prime numbers. Erwin Schrödinger was taught calculus and algebra by Mertens. His memory is honoured by the Franciszek Mertens Scholarship granted to those outstanding pupils of foreign secondary schools who wish to study at the Faculty of Mathematics and Computer Science of the Jagiellonian University in Kraków and were finalists of the national-level mathematics, or computer science olympiads, or they have participated in one of the following international olympiads: in mathematics (IMO), computer science (IOI),

astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galax ...

(IAO), physics (IPhO), linguistics (IOL), or they were participants of the European Girls' Mathematical Olympiad (EGMO).

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* * * {{DEFAULTSORT:Mertens, Franz 1840 births 1927 deaths People from Środa Wielkopolska People from the Province of Posen Polish mathematicians Austro-Hungarian mathematicians Austrian mathematicians 19th-century German mathematicians 20th-century German mathematicians Humboldt University of Berlin alumni Jagiellonian University faculty Academics of the University of Vienna Number theorists

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