Critical line theorem

In mathematics, the critical line theorem tells us that at least a fixed percentage of the nontrivial zeros of the Riemann zeta function, values where $\zeta (s+it)=0$ and $0<s<1$, lie on the critical line where $s={\frac {1}{2}}$. Following work by G. H. Hardy and J. E. Littlewood showing there was an infinity of zeros on the critical line, the theorem was proven for a small percentage by Atle Selberg.

Norman Levinson improved this to one-third of the zeros, and Conrey to two-fifths. The Riemann hypothesis implies that the true value would be one. However, if the true value is one, the Riemann hypothesis is not necessarily implied, because if the zeros not on the critical line are widely-spaced enough, then it is possible that they could comprise "zero percent" of all the zeros within the critical strip.

Bibliography

• Conrey, J. B., More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. reine angew. Math. 399 (1989), 1-16
• Levinson, N., More than one-third of the zeros of Riemann's zeta function are on $\sigma ={\frac {1}{2}}$, Adv. in Math. 13 (1974), 383-436