Critical Line Theorem

Theorem

There exist an infinite number of nontrivial zeroes of the Riemann $\zeta$ function on the critical line.

Proof

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Also see

• Riemann Hypothesis: If this is true, all nontrivial zeroes are on the critical line.

Historical Note

This result was demonstrated by Godfrey Harold Hardy in $1914$.

He and John Edensor Littlewood reiterated the result in $1921$.

Sources

• 1914: G.H. Hardy: Sur les Zéros de la Fonction $\map \zeta s$ de Riemann (C.R. Acad. Sci. Vol. 158: pp. 1012 – 1014)
• 1921: G.H. Hardy and J.E. Littlewood: The zeros of Riemann's zeta-function on the critical line (Math. Z. Vol. 10: pp. 3 – 4)
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,5$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hardy, Godfrey Harold (1877-1947)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hardy, Godfrey Harold (1877-1947)