We examine the behaviour of the zeros of the real and imaginary parts of ξ(s) on the vertical line Rs = 1/2 + λ, for λ 6= 0. This can be rephrased in terms of studying the zeros of families of entire functions A(s) = 1 2 (ξ(s+λ)+ξ(s−λ)) and B(s) = 1 2i (ξ(s+λ)−ξ(s−λ)). We will prove some unconditional analogues of results appearing in [3], specifically that the normalized spacings of the zeros ...

We study the distribution of zeros of holomorphic Hecke cusp forms in several “thin” sets as the weight, k, tends to infinity. We obtain unconditional results for slowly shrinking (with k) hyperbolic balls. This relies on a new, effective, proof of Rudnick’s theorem and on an effective version of Quantum Unique Ergodicity for holomorphic forms, which we obtain in this paper. In addition, assumi...

The zeros of semi-orthogonal functions with respect to a probability measure µ supported on the unit circle can be applied to obtain Szeg˝ o quadrature formulas. The discrete measures generated by these formulas weakly converge to the orthogonality measure µ. In this paper we construct families of semi-orthogonal functions with interlacing zeros, and give a representation of the support of µ in...

We give a definition of the zeros of an infinite-dimensional system with bounded control and observation operators B and C respectively. The zeros are defined in terms of the spectrum of an operator on an invariant subspace. These zeros are shown to be exactly the invariant zeros of the system. For the case of SISO systems, where also the range of B is not in the kernel of C, we show that this ...

An important problem in number theory is to study the distribution of the non-trivial zeros of the Riemann zeta-function which, if one is willing to assume the Riemann Hypothesis, all lie on a vertical line. It is relatively easy to count how many of these zeros lie in a large interval, so the average spacing between consecutive zeros is easy to compute. However, it is a difficult and interesti...

The densities of Yang-Lee zeros for the Ising ferromagnet on the L x L square lattice are evaluated from the exact grand partition functions (L=3 approximately 16). The properties of the density of Yang-Lee zeros are discussed as a function of temperature T and system size L. The three different classes of phase transitions for the Ising ferromagnet--first-order phase transition, second-order p...

The paper presents an $L^{r}-$ analogue of an inequality regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle of arbitrary radius but greater or equal to one. Our result provides improvements and generalizations of some well-known polynomial inequalities.

We study a subtle inequity in the distribution of differences between imaginary parts of zeros of the Riemann zeta function. We establish a precise measure which explains the phenomenon, first observed by R. P. Marco, that the location of each Riemann zero is encoded in the distribution of large Riemann zeros. We also extend these results to zeros of more general L-functions. In particular, we ...

If p(z) is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of p′(z) lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of p(z) to a nearest zero of p′(z)? We obtain bounds for this distance depending on degree. We also show t...

admits a meromorphic continuation to the entire complex plane, with the unique and simple pole of residue 1 at s = 1. In the half-plane {s : Rs ≤ 0}, the Riemann zeta function has simple zeros at −2,−4,−6, . . ., and only at these points which are called trivial zeros. There exist also non-trivial zeros in the band {s : 0 < Rs < 1}. We refer for these basic facts for instance to [Bl] (Propositi...