Mathematics
The Riemann Hypothesis (after Bernhard Riemann) is a conjecture about the zeros of the Riemann's zeta function. It says that all nontrivial zeros of this complex valued function have the real part ½. If the assumption is true or not, is one of the most significant unsolved problems of mathematics.
In 2000, the problem was put by the Clay Mathematics Institute (CMI) on the list of Millennium problems. The institute in Cambridge (Massachusetts) has a prize money of one million dollars for a conclusive solution to the problem in the form of a mathematical proof.
Many famous mathematicians have tried the Riemann Hypothesis. Jacques Hadamard claimed in 1896 without further details
Explanations in his work Sur la distribution of the zéros de la fonction ζ (s) and ses conséquences arithmétiques in which he
Prime number theorem proved that the then recently deceased Stieltjes has proved the Riemann Hypothesis, without the proof
publish. Stieltjes claimed in 1885 in an essay in the Compte Rendu the Académie des Sciences a sentence on the asymptotic
To have proved the behavior of the Mertens function from which the Riemann Hypothesis follows (see below). The famous British
Mathematician Godfrey Harold Hardy used to send a telegram in bad weather before crossing the English Channel
he claimed to have found proof, following the example of Fermat, that on the edge of a book of posterity
He said that he had a proof for his assumption, which unfortunately was too long to find a place on the edge. His colleague John
In Camden in 1906, Edensor Littlewood even got the Riemann Hypothesis as a functional theory problem of his as a student
Professor Ernest William Barnes asked, without connection to the prime number distribution - this relation had Littlewood for itself
discovered and proved in his Fellowship dissertation that the prime number clause follows from the hypothesis, but in continental Europe
was known for some time. As he admitted in his book A mathematician's miscellany, this did not shed any good light on the then state
of mathematics in England. Soon, however, Littlewood made important contributions to analytical number theory
the Riemann Hypothesis. The problem was considered in 1900 by David Hilbert in his list of 23 mathematical problems
Hilbert himself classified it as less difficult than, for example, the Fermat problem: In
In a lecture in 1919 he expressed the hope that a proof would be found during his lifetime, in the case of the Fermat vermination
perhaps in the lifetime of the youngest listener; for the most difficult he considered the transcendental evidence in his problem list -
a problem that was solved by Gelfond and Theodor Schneider in the 1930s. [13] Meanwhile, many of the problems are up
Hilbert's list was resolved, but the Riemann Hypothesis resisted all attempts. Since in the 20th century no proof of the
Riemann's assumption was made, the Clay Mathematics Institute in 2000 again this project to one of
key mathematical problems and exposed a price of one million dollars for conclusive evidence
but not for a counterexample.
There are also analogous conjectures to other Riemann's conjectures for other zeta functions, some of which are well supported numerically
are. In the case of the zeta function of algebraic varieties (the case of the functionary bodies) over the complex numbers, the conjecture in
in the 1930s by Helmut Hasse for elliptic curves and in the 1940s by André Weil for abelian varieties and
proved algebraic curves (even over finite bodies). Because also formulated the Weil conjectures, which include an analogue
belongs to the Riemann hypothesis, for algebraic varieties (also higher dimension than curves) over finite bodies. The proof was
after the development of modern methods of algebraic geometry in the Grothendieck School in the 1970s by Pierre
Deligne provided.
In 1945, Hans Rademacher claimed to have refuted the assumption, causing quite a stir in the United States. Just before the
But publication in the Transactions of the American Mathematical Society Carl Ludwig Siegel still found a mistake. Alan
Turing was also of the opinion that the assumption was wrong. He dealt intensively with the calculation of zeros of the
Zeta function and tried to build a mechanical machine shortly before his involvement in deciphering work at Bletchley Park
To help him find at least one guess that violates (and thus refutes) the guess.
Louis de Branges de Bourcia spent decades working on the problem. 1985 (shortly after his proof of the Bieberbach conjecture)
He presented a proof based on his theory of Hilbert dreams of whole functions, in which Peter Sarnak a
Error found. In 1989, when presenting a series of lectures at the Institut Henri Poincaré, he presented further evidence, which he soon afterwards
even recognized as faulty. In 2004 he published a new proof, which was critically examined. Eberhard had been there years before
On Friday, however, a counterexample was made of an allegation made in the proof, so that the evidence is now considered false
becomes.
