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): $0,5$