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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Differentiation evens out zero spacings

Authors: David W. Farmer and Robert C. Rhoades
Journal: Trans. Amer. Math. Soc. 357 (2005), 3789-3811
MSC (2000): Primary 30C15
Published electronically: March 31, 2005
MathSciNet review: 2146650
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Abstract: If $f$ is a polynomial with all of its roots on the real line, then the roots of the derivative $f'$ are more evenly spaced than the roots of $f$. The same holds for a real entire function of order 1 with all its zeros on a line. In particular, we show that if $f$ is entire of order 1 and has sufficient regularity in its zero spacing, then under repeated differentiation the function approaches, after normalization, the cosine function. We also study polynomials with all their zeros on a circle, and we find a close analogy between the two situations. This sheds light on the spacing between zeros of the Riemann zeta-function and its connection to random matrix polynomials.

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Additional Information

David W. Farmer
Affiliation: American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94306-2244

Robert C. Rhoades
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Received by editor(s): October 21, 2003
Received by editor(s) in revised form: March 25, 2004
Published electronically: March 31, 2005
Additional Notes: Research of the first author was supported by the American Institute of Mathematics and the NSF
Article copyright: © Copyright 2005 American Mathematical Society

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