We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension n having m singular points. As a function of n, m, this bound turns out to be double exponential in the prec...

In this paper, we present a correct proof of an L p-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.

It is proved that a polynomial p of the form

0. Introduction In a remarkable numerical experiment, Odlyzko Od] has found that the local spacing distribution between the zeros of the Riemann Zeta function is modelled by the eigen-value distributions coming from random matrix theory. In particular by the \GUE" (Gaussian Unitary Ensemble) model Gau]. His experiment was inspired by the paper of Montgomery Mon] who determined the pair correlat...