# 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

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## 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

- April 1974: Norman Levinson:
*At least one third of the zeros of Riemann's zeta-function are on $\sigma = 1/2$*(*Proc. Natl. Acad. Sci. U S A***Vol. 71**: pp. 1013 – 1015) www.jstor.org/stable/63251

- 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $0,33333 33333 33 \ldots$ - 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $0,5$