Editing Riemann zeta function (section)

=== Other results ===
It is known that there are infinitely many zeros on the critical line. [[John Edensor Littlewood|Littlewood]] showed that if the sequence ({{math|''γ<sub>n</sub>''}}) contains the imaginary parts of all zeros in the [[upper half-plane]] in ascending order, then

:<math>\lim_{n\rightarrow\infty}\left(\gamma_{n+1}-\gamma_n\right)=0.</math>

The [[critical line theorem]] asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is 1.)

In the critical strip, the zero with smallest non-negative imaginary part is {{math|{{sfrac|1|2}} + 14.13472514...''i''}} ({{OEIS2C|A058303}}). The fact that
:<math>\zeta(s)=\overline{\zeta(\overline{s})}</math>
for all complex {{math|''s'' ≠ 1}} implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line {{math|Re(''s'') {{=}} {{sfrac|1|2}}}}.

It is also known that no zeros lie on the line with real part 1.