Differentiation evens out zero spacings |
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Authors: | David W Farmer Robert C Rhoades |
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Institution: | American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94306-2244 ; Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837 |
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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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Keywords: | |
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