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ISSN 1088-6850(e) ISSN 0002-9947(p)

Differentiation evens out zero spacings

Author(s): David W. Farmer; Robert C. Rhoades
Journal: Trans. Amer. Math. Soc. 357 (2005), 3789-3811.
MSC (2000): Primary 30C15
Posted: March 31, 2005
MathSciNet review: 2146650
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Abstract: If is a polynomial with all of its roots on the real line, then the roots of the derivative are more evenly spaced than the roots of . The same holds for a real entire function of order 1 with all its zeros on a line. In particular, we show that if 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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