Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 30C15

Retrieve articles in all journals with MSC (2000): 30C15


Additional Information

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

Robert C. Rhoades
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
Email: rrhoades@bucknell.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03721-9
PII: S 0002-9947(05)03721-9
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