who Mertens was?
Mertens is the well-known number theorist Franciszek Mertens who was born
on March 20, 1840, in Schroda, Posen (a former Prussian province, now Sroda,
in Poland), and died in Vienna, on March 5, 1927.
As we all know, Gauss died in 1855, and that very year Dirichlet, who had
been teaching at the University of Berlin since 1828, went to Go"ttingen
to succeed Gauss. Therefore, Dirichlet's chair became vacant in 1855, and
Kummer came to Berlin to be appointed professor, a position he held until
he retired in 1883. As we also know, Kronecker was a student of Kummer, and
it seems apparent that it was due to him that Kronecker became interested in
number theory.
In his youth, Mertens moved to Berlin where he became a student at Berlin
University, and where he studied under Kronecker and Kummer.
I find somehow unfair to state, as I've read, that "Mertens was a number
theorist who is best remembered for his elementary proof of the Dirichlet
theorem which appears in most modern textbooks". As a matter of fact, Franz
Mertens published a long string of important papers on many topics.
For instance, this is the very Mertens who, in 1886, re-proved Gordan's
theorem for binary systems by an inductive method.
According to Morris Kline [MT from A to MT, p.929], Mertens "assumed the
theorem to be true for any given set of binary forms and then proved it must
still be true when the degree of one of the forms is increased by one. He
did not exhibit explicitly the finite set of independent invariants and
covariants but he proved that it existed. The simplest case, a linear form,
was the starting point of the induction and such a form has only powers of
itself as covariants.
Hilbert, after writing a doctoral thesis in 1885 on invariants, in 1888 also
re-proved Gordan's theorem that any given system of binary forms has a finite
complete system of invariants and covariants. His proof was a modification
of Mertens's. Both proofs were far simpler than Gordan's. But Hilbert's proof
also did not present a process for finding the complete system.
...
Hilbert's existence proof was so much simpler than Gordan's laborious
calculation of a basis that Gordan could not help exclaiming, 'This is not
mathematics; it is theology'. However, he reconsidered the matter and said
later, 'I have convinced myself that theology also has its advantages'.
In fact he himself simplified Hilbert's existence proof." [1]
As an anecdote, I recall that on October 21, 1881, Heine died, and a
replacement was needed to fill the chair at Halle. Cantor thought of three
mathematicians for the new available position. Dedekind was first in Cantor's
list, followed by Heinrich Weber, and Franz Mertens. But, as it seems, in
1882 Cantor got surprised when Dedekind declined the offer, and the shock
was even worse when Weber, and also Mertens declined.
Franz Mertens first worked in Cracow, and then moved to Austria. Incidentally,
Ernst Fischer and Schro"dinger, for instance, were students of Mertens at the
University of Vienna.
[1] Nachrichten Ko"nig. Gesellschaft der Wissenschaften zu Go"ttingen, 1899,
pp 240-242.
[2] Dick, Auguste:
"Franz Mertens: 1840-1927: eine biographische Studie", mit einer Einl. von
Edmund Hlawka, Graz: Berichte der mathematisch-statistischen Sektion im
Forschungszentrum Nr. 151, 38 pages, 1981.
[3] te Riele, Herman J.J.:
"Some Historical and Other Notes about the Mertens Conjecture and its
Recent Disproof, _Nieuw Archief voor Wiskunde_ Vierde Serie, vol 3, no 2,
pp 237-243, 1985.
This article answers some questions about the disproof of the Mertens
conjecture by A.M. Odlyzko and H.J.J. te Riele [cf. "Disproof of the
Mertens conjecture", _J. Reine Angew. Math._ vol 357, pp 138-160, 1985].
This conjecture (1897) attracted lots of interest in its almost 100 years
of existence because its truth would imply the truth of the Riemann
hypothesis. The disproof relies on extensive computations with the zeros
of the zeta-function, and does not provide an explicit counterexample. In
this paper [2], the roles of Stieltjes and Mertens are sketched in their
historical perspective.
[4] Domoradzki, Stanislaw:
"Franciszek Mertens (1840-1927)" (in Polish. English summary), _Opuscula
Mathematica_ vol 13, pp 109-115, 1993.
posted on Historia Matematica Mailing List by Julio Gonzalez Cabillon (jgc@adinet.com.uy) on Thu, 23 Sep 1999
