Characteristic polynomials of unitary matrices are extremely useful models for the Riemann zeta-function ζ(s). The distribution of their eigenvalues give insight into the distribution of zeros of the Riemann zeta-function and the values of these characteristic polynomials give a model for the value distribution of ζ(s). See the works [KS] and [CFKRS] for detailed descriptions of how these model...

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity. Mathematics Subject Classification (2010). Primary: 11F11, 11F30. Se...

Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t, . . . , tn) projected onto the surface of the unit sphere, divided by π. The probability density...

the aim of this paper is to classify the finite simple groups with the number of zeros at most seven greater than the number of nonlinear irreducible characters in the character tables. we find that they are exactly a$_{5}$, l$_{2}(7)$ and a$_{6}$.

We study so-called real zeros of holomorphic Hecke cusp forms, that is zeros on three geodesic segments on which the cusp form (or a multiple of it) takes real values. Ghosh and Sarnak, who were the first to study this problem, showed that existence of many such zeros follows if many short intervals contain numbers whose all prime factors belong to a certain subset of the primes. We prove new r...

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.