Editing Analytic number theory (section)

===Riemann===
{{main|Bernhard Riemann}}
{{quote box
| align = right
| width = 30%
| quote = "{{lang|de|…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.}}"<br /><br />"…it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the next objective of my investigation."
| source = Riemann's statement of the Riemann hypothesis, from his 1859 paper.<ref name="Riemann1859">{{citation|first=Bernhard |last=Riemann |author-link=Bernhard Riemann |url=http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/ |title={{sic|hide=y|Ueber}} die Anzahl der Primzahlen unter einer gegebenen {{sic|hide=y|Grösse}} |year=1859 |journal=Monatsberichte der Berliner Akademie }}. In ''Gesammelte Werke'', Teubner, Leipzig (1892), Reprinted by Dover, New York (1953). [http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/ Original manuscript] {{webarchive |url=https://web.archive.org/web/20130523061451/http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/ |date=May 23, 2013 }} (with English translation). Reprinted in {{harv|Borwein|Choi|Rooney|Weirathmueller|2008}} and {{harv|Edwards|1974}}</ref> (He was discussing a version of the zeta function, modified so that its roots are real rather than on the critical line. See, Riemann Xi Function.)
}}

[[Bernhard Riemann]] made some famous contributions to modern analytic number theory. In [[On the Number of Primes Less Than a Given Magnitude|a single short paper]] (the only one he published on the subject of number theory), he investigated the [[Riemann zeta function]] and established its importance for understanding the distribution of [[prime numbers]]. He made a series of conjectures about properties of the [[Riemann zeta function|zeta function]], one of which is the well-known [[Riemann hypothesis]].