There is an intimate connection between the distribution of the nontrivial zeros of the Riemann zeta function ζ(s) and the distribution of prime numbers. Critical to many prime number problems is the horizontal distribution of zeros; here the Riemann Hypothesis (RH) asserts that the zeros all have real part 1 2 . There is also much interest in studying the distribution of the imaginary parts of...

We define a “renormalized energy” as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of [SS1, SS3]. Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number of charges. The functional is expected to be a good measure of disorder of a...

We prove a complex analytic analogue of the classical Rolle theorem asserting that the number of zeros of a real smooth function can exceed that of its derivative by at most 1. This result is used then to obtain upper bounds for the number of complex isolated zeros of : (1) functions defined by linear ordinary differential equations (in terms of the magnitude of the coefficients of the equation...

The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree N in an open set U ⊂ C as the degree N → ∞, and more generally for the zeros of random holomorphic sections of high powers of any positive line bundle over any Riemann surface. The formulas were conjectured in special cases by Forrester and Honner. In higher...

Of central importance in number theory is the distribution of the complex zeros of f(s), all of which are in the critical strip 0 < a < 1 and are symmetrically located about the real axis and about the critical line a = 1/2. Riemann conjectured in 1859 that all of these zeros are on the critical line; this conjecture, which is still unproved, is known as the Riemann Hypothesis. The number of ze...

Zeros of Kac polynomials spanned by monomials with i.i.d. random coefficients are asymptotically uniformly distributed near the unit circumference. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by different bases, e.g., by orthogonal...