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Three Circles Theorem


The three circles theorem, also called Hadamard's three circles theorem (Edwards 2001, p. 187), states that if f is an analytic function in the annulus 0<r_1<|z|<r_2<infty, r_1<r<r_2, and M_1, M_2, and M are the maxima of f on the three circles corresponding to r_1, r_2, and r, respectively, then

 M^(ln(r_2/r_1))<=M_1^(ln(r_2/r))M_2^(ln(r/r_1))

(Derbyshire 2004, p. 376).

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


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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 zeta(s) n'a pas de zéros dans le demi-plan R(x)>1/2." 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.

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Three Circles Theorem

Cite this as:

Weisstein, Eric W. "Three Circles Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ThreeCirclesTheorem.html

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