Person: Mertens, Franz Carl Joseph
Franz Mertens was a Polish-born mathematician who made contributions to a wide variety of areas. He formulated the Mertens conjecture which (had it been true) would have implied the Riemann Hypothesis.
Mathematical Profile (Excerpt):
- Sroda, the village in which Mertens was born, was close to Poznań in Prussia.
- He was known, however, both by the German version of his name which is Franz and the Polish version which is Franciszek.
- Mertens completed his university studies at the University of Berlin where he attended lectures by Weierstrass, Kronecker and Kummer.
- This was the "golden period" for mathematics in Berlin and gave Mertens the best possible mathematical foundations.
- It is worth noting that, in 1882, Mertens was offered the chair at Halle left vacant after the death of Heine.
- He had been the third choice for this chair but after Dedekind and Heinrich Weber turned it down, Mertens (who was at that time in Cracow) also turned it down.
- Mertens worked on a number of different topics including potential theory, geometrical applications to determinants, algebra and analytic number theory, publishing 126 papers.
- Many people are aware of Mertens contributions since his elementary proof of the Dirichlet theorem appears in most modern textbooks.
- However he made many deep contributions including Mertens' theorems, three results in number theory related to the density of the primes.
- Here Mertens defines M(n)M(n)M(n) to be the sum of the numbers m(i)m(i)m(i) where iii runs from 1 to nnn and where mmm is the Möbius function.
- It was known that there is no xxx with M(x)>xM(x) > xM(x)>x but Mertens' conjecture was stronger, namely that there is no xxx with M(x)>√xM(x) > √xM(x)>√x.
- The result is important since a proof of Mertens' conjecture would imply the truth of the Riemann hypothesis.
- Among Mertens other papers we mention: Invariante Gebilde ternärer Formen Ⓣ(Invariant structure of ternary forms) (1887); Invariante Gebilde quaternärer Formen Ⓣ(Invariant structure of quaternary forms) (1889); Dirichletscher Reihen Ⓣ(Dirichlet series) (1895); Zur linearen Transformation der q-Reihen Ⓣ(On linear transformation of q-series) (1901); and Beweis der Galois'schen Fundamentalsatzes Ⓣ(Proof of Galois's Fundamental Theorem) (1902).
Born 20 March 1840, Schroda, Posen, Prussia (now Środa Wielkopolska, Poland). Died 5 March 1927, Vienna, Austria.
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Adapted from other CC BY-SA 4.0 Sources:
- O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive