Tail bounds for counts of zeros and eigenvalues, and an application to ratios

  • Brad Rodgers

    University of Zürich, Switzerland

Abstract

Let be random and uniformly distributed in the interval , and consider the quantity , a count of zeros of the Riemann zeta function in a box of height . Conditioned on the Riemann hypothesis, we show that the probability this count is greater than decays at least as quickly as , uniformly in . We also prove a similar results for the logarithmic derivative of the zeta function, and likewise analogous results for the eigenvalues of a random unitary matrix.

We use results of this sort to show on the Riemann hypothesis that the averages [ \frac{1}{T} \int_T^{2T} \Bigg| \frac{\zeta\big(\frac{1}{2} + \frac{\alpha}{\log T} + it\big)}{\zeta\big(\frac{1}{2}+ \frac{\beta}{\log T} + it\big)}\Bigg|^m,dt ] remain bounded as , for complex numbers with . Moreover we show rigorously that the local distribution of zeros asymptotically controls ratio averages like the above; that is, the GUE Conjecture implies a (first-order) ratio conjecture.

Cite this article

Brad Rodgers, Tail bounds for counts of zeros and eigenvalues, and an application to ratios. Comment. Math. Helv. 92 (2017), no. 2, pp. 311–347

DOI 10.4171/CMH/413