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).
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. Paris154,
263-266, 1912.Robinson, R. M. "Hadamard's Three Circles Theorem."
Bull. Amer. Math. Soc.50, 795-802, 1944.
is an analytic function
in the annulus 
,
,
and 
,
,
and 
are the maxima of 
on the three circles corresponding to 
, 
, and 
, respectively, then
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.