ON THE NUMBER OF PRIMES LESS THAN A GIVEN MAGNITUDE

'''Über die Anzahl der Primzahlen unter einer gegebenen Größe''' (Usual English translation: '''On the Number of Primes Less Than a Given Magnitude''') is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the ''Monatsberichte der Königlich Preußischen Akadademie der Wissenschaften zu Berlin''. Although it is the only paper he ever published on number theory, it contains ideas which influenced thousands of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory. The paper was so influential that the notation $s\; =\; sigma\; +\; i\; t$ is used to denote a complex number while discussing the zeta function (see below) instead of the usual ''z=x+iy''.
Among the new definitions introduced:

★ The use of the Greek letter zeta (ζ) for a function previously mentioned by Euler

★ The analytic continuation of this zeta function ζ(''s'') to all complex ''s'' ≠ 1

★ The entire function ξ(''s''), related to the zeta function through the gamma function (or the Π function, in Riemann's usage)

★ The discrete function ''J''(''x'') defined for ''x'' ≥ 0, which is defined by ''J''(0) = 0 and ''J''(''x'') jumps by 1/''n'' at each prime power ''p''^''n''. (Riemann calls this function f(x).)
Among the proofs and sketches of proofs:

★ Two proofs of the functional equation of ζ(''s'')

★ "Proof" of the product representation of ξ(''s'')

★ "Proof" of the approximation of the number of roots of ξ(''s'') whose imaginary part lies between 0 and ''T''
Among conjectures made:

★ The Riemann hypothesis, that all (nontrivial) zeros of ζ(''s'') have real part 1/2. Riemann states this in terms of the roots of the related ξ function, "...es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien." That is, "it is very probable that all roots are real. Of course one would wish for a stricter proof here; I have for the time being, after some fleeting futile attempts, provisionally put aside the search for this, as it appears unnecessary for the next objective of my investigation."
New methods and techniques used in number theory:

★ Analytic continuation (although not in the spirit of Weierstrass)

★ Contour integration

★ Fourier inversion
Riemann also discussed the relationship between ζ(''s'') and the distribution of the prime numbers, using the function ''J''(''x'') essentially as a measure for Stieltjes integration. He then obtained the main result of the paper, a formula for ''J''(''x''), by comparing with ln(ζ(''s'')). Riemann then found a formula for the prime-counting function π(''x'') (which he calls ''F''(''x'')). He notes that his equation explains the fact that π(''x'') grows more slowly than the logarithmic integral, as had been found by Gauss and a certain Goldschmidt.
The paper contains some peculiarities for modern readers, such as the use of Π(''s''-1) instead of Γ(''s''), or writing ''tt'' instead of ''t''². The style can also be surprising, such as writing an integral from $infty$ to $infty$.

Contents

External links

References

External links

★ English translation of Riemann's paper

★ Number theory and physics website

★ Riemann's article in original German

★ Photo of Riemann's (almost illegible) original manuscript

References

★ ''Riemann's Zeta Function'', H. M. Edwards, Dover, 1974, ISBN 0-486-41740-9