Critical line theorem

Critical line theorem

In mathematics, the critical line theorem says that a positive proportion of the nontrivial zeros of the Riemann zeta function lie on the critical line. Following work by harvs|authorlink=G. H. Hardy|first=G. H. |last=Hardy|year=1914|txt=yes and harvs|last=Hardy|last2=Littlewood|author2-link=J. E. Littlewood|year=1921|txt=yes showing there was an infinity of zeros on the critical line, the theorem was proven for a small positive proportion by harvs|first=Atle|last= Selberg|authorlink=Atle Selberg|year=1942|txt=yes.

harvs|authorlink=Norman Levinson|last=Levinson|first=Norman|year=1974|txt=yes improved this to one-third of the zeros, and harvtxt|Conrey|1989 to two-fifths. The Riemann hypothesis implies that the true value would be one.

References

*citation|id=MR|1004130 | last=Conrey|first= J. B.|title=More than two fifths of the zeros of the Riemann zeta function are on the critical line|journal= J. Reine angew. Math.|volume= 399 |year=1989|pages= 1–16
url=http://www.digizeitschriften.de/resolveppn/GDZPPN002206781
*citation|last=Hardy|first= G. H.|title= Sur les Zéros de la Fonction ζ(s) de Riemann|journal=C. R. Acad. Sci. Paris |volume=158|pages= 1012–1014 |year=1914
*citation|last=Hardy|first= G. H.|last2= Littlewood|first2= J. E.|title= The zeros of Riemann's zeta-function on the critical line|journal= Math. Z.|volume= 10|pages= 283–317 |year=1921|doi=10.1007/BF01211614
*citation|id=MR|0564081|last=Levinson|first=N.|title= More than one-third of the zeros of Riemann's zeta function are on σ = 1/2|journal= Adv. In Math. |volume=13 |year=1974|pages= 383–436
doi=10.1016/0001-8708(74)90074-7
*citation|id=MR|0010712
last=Selberg|first= Atle
title=On the zeros of Riemann's zeta-function.
journal=Skr. Norske Vid. Akad. Oslo I. |year=1942|volume= 10|pages= 59 pp


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