Bounds on the number of simple zeros of the derivatives of a function are used to give bounds on the number of distinct zeros of the function. The Riemann ξ-function is defined by ξ(s) = H(s)ζ(s), where H(s) = 2s(s−1)π 1 2 Γ( 1 2 s) and ζ(s) is the Riemann ζ-function. The zeros of ξ(s) and its derivatives are all located in the critical strip 0 < σ < 1, where s = σ+ it. Since H(s) is regular an...

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...

In general, the zeros of an orthogonal rational function (ORF) on a subset of the real line, with poles among {α1, . . . , αn} ⊂ (C0 ∪ {∞}), are not all real (unless αn is real), and hence, they are not suitable to construct a rational Gaussian quadrature rule (RGQ). For this reason, the zeros of a so-called quasi-ORF or a so-called paraORF are used instead. These zeros depend on one single par...

Abstract. The complex zeros of the Riemannn zeta-function are identical to the zeros of the Riemann xi-function, ξ(s). Thus, if the Riemann Hypothesis is true for the zetafunction, it is true for ξ(s). Since ξ(s) is entire, the zeros of ξ(s), its derivative, would then also satisfy a Riemann Hypothesis. We investigate the pair correlation function of the zeros of ξ(s) under the assumption that ...

Unstable zeros limit the achievable control performance. When a continuous-time system is discretized using the zero-order hold, there is no simple relation which shows how the zeros of the continuous-time system are transformed by sampling. This paper analyzes the asymptotic behavior of the limiting zeros for multivariable systems and derives a new condition for the zeros to be stable for suff...

Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an elementary combinatorial argument, we prove that assuming the Riemann Hypothesis at least 67.275% of the zeros are simple.

The purpose of this paper is to present three new methods for finding all simple zeros of polynomials simultaneously. First, we give a new method for finding simultaneously all simple zeros of polynomials constructed by applying the Weierstrass method to the zero in the trapezoidal Newton’s method, and prove the convergence of the method. We also present two modified Newton’s methods combined w...