Differentiation evens out zero spacings
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- by David W. Farmer and Robert C. Rhoades
- Trans. Amer. Math. Soc. 357 (2005), 3789-3811
- DOI: https://doi.org/10.1090/S0002-9947-05-03721-9
- Published electronically: March 31, 2005
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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.References
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Bibliographic Information
- David W. Farmer
- Affiliation: American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94306-2244
- MR Author ID: 341467
- Email: farmer@aimath.org
- Robert C. Rhoades
- Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
- MR Author ID: 762187
- Email: rrhoades@bucknell.edu
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 3789-3811
- MSC (2000): Primary 30C15
- DOI: https://doi.org/10.1090/S0002-9947-05-03721-9
- MathSciNet review: 2146650