Three Circles Theorem

The three circles theorem, also called Hadamard's three circles theorem (Edwards 2001, p. 187), states that if  is an analytic function in the annulus , , and , , and  are the maxima of  on the three circles corresponding to , , and , respectively, then

(Derbyshire 2004, p. 376).

The theorem was first published by Hadamard in 1896, although without proof (Bohr and Landau 1913; Edwards 2001, p. 187).

REFERENCES:

Bohr, H. and Landau, E. "Beiträge zur Theorie der Riemannschen Zetafunktion." Math. Ann. 74, 3-30, 1913. Reprinted in Bohr, H. §B11 in Collected Works, Vol. 1.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, pp. 159 and 376, 2004.

Edwards, H. M. "The Three Circles Theorem." §9.3 in Riemann's Zeta Function. New York: Dover, pp. 187-188, 2001.

Littlewood, J. E. "Quelques conséquences de l'hypothèse que la fonction  n'a pas de zéros dans le demi-plan ." C. R. Acad. Sci. Paris 154, 263-266, 1912.

Robinson, R. M. "Hadamard's Three Circles Theorem." Bull. Amer. Math. Soc. 50, 795-802, 1944.