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 G. H. Hardy (1914) and Hardy and Littlewood (1921) showing there was an infinity of zeros on the critical line, the theorem was proven for a small positive proportion by Atle Selberg (1942).

Norman Levinson (1974) improved this to one-third of the zeros, and Conrey (1989) to two-fifths. The Riemann hypothesis implies that the true value would be one.

[edit] References

Conrey, J. B. (1989), “More than two fifths of the zeros of the Riemann zeta function are on the critical line”, J. reine angew. Math. 399: 1-16, MR 1004130 , <http://www.digizeitschriften.de/resolveppn/GDZPPN002206781>
Hardy, G. H. (1914), “Sur les Zéros de la Fonction ζ(s) de Riemann”, C. R. Acad. Sci. Paris 158: 1012-1014
Hardy, G. H. & Littlewood, J. E. (1921), “The zeros of Riemann's zeta-function on the critical line”, Math. Z. 10: 283-317, DOI 10.1007/BF01211614
Levinson, N. (1974), “More than one-third of the zeros of Riemann's zeta function are on σ = 1/2”, Adv. in Math. 13: 383-436, MR 0564081, DOI 10.1016/0001-8708(74)90074-7
Selberg, Atle (1942), “On the zeros of Riemann's zeta-function.”, Skr. Norske Vid. Akad. Oslo I. 10: 59 pp, MR 0010712