At Least One Third of Zeros of Riemann Zeta Function on Critical Line

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Theorem

At least $\dfrac 1 3$ of the nontrivial zeroes of the Riemann $\zeta$ function lie on the critical line.


Proof


Historical Note

This result, published in April $1974$, was one of the last things Norman Levinson demonstrated.

In August $1974$ he published a further result: that the inequality is in fact strict.

In $1975$ he died of a brain tumour.


The result has since been improved, and it is now known that at least $\dfrac 2 5$ of the nontrivial zeroes of the Riemann zeta function lie on the critical line.


Sources