We study the distribution of the nontrivial zeros of ideal class zeta functions associated to elements in the symmetric space of GLn over a number field. We establish asymptotics for the number of nontrivial zeros up to height T , and asymptotics for the distribution of the nontrivial zeros with respect to the critical line. We combine these results to study the mean value of the real parts of ...

Let f be a real entire function with finitely many non-real zeros, not of the form f = Ph with P a polynomial and h in the Laguerre-Pólya class. Lower bounds are given for the number of non-real zeros of f ′′ + ωf , where ω is a positive real constant.

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the n-th derivative tends to infinity as n → ∞ . We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.

We study the average number of intersecting points of a given curve with random hyperplanes in an n-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan this problem is closely linked to nding the average number of real zeros of random polynomials. They show that a real polynomial of degree n has in average 2 log n + O(1) real zeros (M. Kac's theorem). This result leads us to th...

We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree n, with not necessarily independent coefficients, decays like √ logn/n. Our proofs rely on discrepancy results for deterministic polynomials, and order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of t...

This paper provides asymptotic estimates for the expected number of real zeros of two different forms of random trigonometric polynomials, where the coefficients of polynomials are normally distributed random variables with different means and variances. For the polynomials in the form of a0 a1 cos θ a2 cos 2θ · · · an cosnθ and a0 a1 cos θ b1 sin θ a2 cos 2θ b2 sin 2θ · · · an cosnθ bn sinnθ,w...