Franz Mertens (20 March 1840 – 5 March 1927) (also known as Franciszek Mertens) was a Polish

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. 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 The Kingdom of Prussia (german: Königreich Preußen, ) was a German kingdom that constituted the state of Prussia between 1701 and 1918. Marriott, J. A. R., and Charles Grant Robertson. ''The Evolution of Prussia, the Making of an Empire''. ...

(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 Count ...

Poland Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populou ...

) and died in

Vienna en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST ...

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 ...

. The

Mertens function In number theory, the Mertens function is defined for all positive integers ''n'' as : M(n) = \sum_^n \mu(k), where \mu(k) is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive re ...

''M''(''x'') is the sum function for the

Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most of ...

, 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 d ...

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 188 ...

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 ...

is analogous to the

Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...

, 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 Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...

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 Kraków (), or Cracow, is the second-largest and one of the oldest cities in Poland. Situated on the Vistula River in Lesser Poland Voivodeship, the city dates back to the seventh century. Kraków was the official capital of Poland until 1596 ...

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 Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...

(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, g ...

(IAO),

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

(IPhO),

linguistics Linguistics is the science, scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure ...

(IOL), or they were participants of the

European Girls' Mathematical Olympiad The European Girls' Mathematical Olympiad (EGMO) is a mathematical olympiad for girls which started in 2012. It is similar to, and was inspired by, the China Girls Mathematical Olympiad (CGMO). Although the competition is held in Europe, several co ...

(EGMO).

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External links

* * * {{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

See also

References

External links