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Riemann zeta function
(section)
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=== 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|''Ξ³<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'' β 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.
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