Differentiation Evens Out Zero Spacings

Details Differentiation Evens Out Zero Spacings Differentiation Evens Out Zero Spacings

2003-10-17

arxiv.org/abs/math/0310252v3

Mathematics

Number Theory

22 pages, 1 figure. Final version. To appear in TAMS

Scientific paper

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 (a change of variables from) 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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