On the number of prime numbers lower than a given size

On the number of prime numbers lower than a given size (gold Über die Anzahl der Primzahlen unter einer gegebenen Grösse ) is an article of 8 pages writes by Bernhard Riemann and published in the edition of November 1859 of the Monthly returns of the Academy of Berlin . Although it is the only article which it published on the Théorie of the numbers, it contains ideas which influenced thousands of researchers since the end of the 19th century until our days.

Description

The article contains initially heuristic definitions, arguments, drafts of evidence and the application of powerful analytical methods; all those became essential concepts and tools of the analytical Théorie of the numbers modern.

Among the new introduced definitions:

the analytical Prolongation of the Function zeta of Riemann $\ zeta (S) \,$ with all the complex numbers $s$ different from 1;
the whole Function $\ xi (S)$ ;
the discrete function J ( X ) definite for $x \ Ge 0 \,$ , which is defined by J (0) = 0 and J ( X ), change by jumps of $\ frac {1} {N} \,$ with each power of prime number $p^n \,$ .

Among the evidence and the drafts of evidence:

two evidence of the functional equation of $\ zeta (S)$ ;
the proof of the representation by product of $\ xi (S)$ ;
the proof of the approximation of the number of roots of $\ xi (S) \,$ of which the imaginary parts are located between 0 and T .

Among the produced conjectures:

the Assumption of Riemann, all the zeros (noncommonplace) of $\ zeta (S) \,$ have a real part equalizes with $\ frac 1 2$ .

New methods and techniques used in theory of the numbers:

analytical Prolongation (different from the direction of Weierstrass);
Integral curvilinear;
Inversion of Fourier.

Riemann also discussed of the relations between $\ zeta (S) \,$ and the distribution of the prime numbers, using the function J ( X ) primarily like a measurement for the integration of Stieltjes. It then obtained the principal result of the article, a formula for J ( X ), by comparison with $\ ln (\ zeta (S) \,$ ). Riemann then tried to work out an approximate formula for the Fonction of account of the prime numbers $\ pi (X) \,$ , although he admitted itself to be conscious of the defect of his arguments.

Anecdote

The article had such an influence that the notation

s = \ sigma + I T \,

is used to note a Complex number at the time of the study of the function zeta in the place of the usual notation

z=x+iy \,

See too

External bonds

Article of Riemann
Number theory and physics website

References

H. Mr. Edwards, Riemann' S Zeta Function , Dover, 1974, ISBN 0486417409