Assumption of Riemann

The assumption of Riemann is a Conjecture formulated in 1859 by the mathematician Bernhard Riemann. She says that the zero noncommonplace of the Fonction zeta of Riemann have all to some extent real 1/2. Its demonstration would improve knowledge of the distribution of the prime numbers.

This conjecture constitutes one of the unsolved problems most important of mathematics of the beginning of XXIe century: it is one of the famous Problèmes of Hilbert suggested in 1900, and is the subject of one of the problems Clay for the third millennium, equipped with a price of a million USD.

History

Riemann mentioned the conjecture, called later “assumption of Riemann”, in its article published in 1859 Concerning the number of prime numbers lower than a given size ( Über die Anzahl der Primzahlen unter einer gegebenen Grösse ), but this conjecture not being the principal subject of its article, it does not await a demonstration. Riemann knew that zero the noncommonplace ones of the function zeta were distributed symmetrically around the line $s\; =\; \backslash \; frac\; \{1\}\; \{2\}\; +\; it\; \backslash ,$ and knew that all zero the noncommonplace ones were in the band $0\; \backslash \; Re\; (S)\; \backslash \; the\; 1$.

In 1896, Hadamard and of the Valley-Chick independently proved that no zero could be on the line $Re\; (S)\; =\; 1\; \backslash ,$, as all zero the noncommonplace ones were to be in the interior of the band criticizes . This was a result-key in the first complete demonstration of the Théorème of the prime numbers.

In 1900, Hilbert includes the assumption of Riemann in its famous list of 23 unsolved problems - it belongs to the 8 {{E}} problem, this one also including/understanding the Conjecture of Goldbach. He would have said in connection with this problem: “If I were to awake me after having slept during thousand years, my first question would be: Was the assumption of Riemann proven? ” . The assumption of Riemann is the only problem of Hilbert to be appeared in the list of the Problèmes of the price of the millenium of the Institut of mathematics Clay.

In 1914, Hardy proved that there is an infinity of zeros on the line criticizes $Re\; (S)\; =\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash ,$. However there remains possible that there is an infinity of zero noncommonplace elsewhere. later work of Hardy and Littlewood in 1921 and of Selberg in 1942 gave an estimate of the average density of zeros on the critical line.

Recent work was focused on the explicit calculation of places where many of zeros are (in the hope to find a counterexample) and to place hight delimiters on the proportion of zeros being elsewhere than on the critical line (in the hope to reduce it to zero).

Numerical tests

In the absence of demonstration validated by the community of the mathematicians, Andrew Mr. Odlyzko specialized in numerical calculation of zero the noncommonplace ones of the function. It is thus generally affirmed that the calculated billion and half of zeros check all the assumption of Riemann; what means that they are positioned enough close critical line (with the direction which the inaccuracy of calculation is such as they can be there exactly).

Tests of demonstration

A many supposed evidence of the assumption of Riemann is regularly proposed, mainly on Internet, like some invalidations, often the fact of mathematicians in margin of the traditional university system. None of this work has for the moment received the approval of the mathematical community.

The site counts some of this supposed evidence (including “evidence” that the assumption would be false), and also some parodies.

See too

References

Random links:

Sorquainville | Antoine Elie Lamblardie | The Report/ratio | Hugues III of Rethel | Olaf Scholz

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