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...

In 1972 Montgomery [20, 21] introduced a new method for studying the zeros of the Riemann zeta-function. One of his main accomplishments was to determine partially the pair correlation of zeros, and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros. Perhaps more importantly, he conjectured on number-theoretic grounds an asymptotic formula for the pai...

It is well known that multirate input and hold, such as a generalized sample hold function (GSHF), can be used to shift the sampling zeros of a discrete-time model for a continuous-time system. This paper deals with the stability of sampling zeros, as the sampling period tends to zero, of discrete-time models that are composed of a GSHF, a continuous-time multivariable plant and a sampler in ca...

with s = 12 + it , and shows that ξ(t) is an even entire function of t whose zeros have imaginary part between −i/2 and i/2. He further states, sketching the proof, that in the range between 0 and T the function ξ(t) has about (T/2π) log(T/2π)− T/2π zeros. Riemann then continues: “Mann finden nun in der That etwa so wiel reelle Nullstellen innerhalb dieser Grenzen, und es ist sehr wahrscheinlic...

The Riemann hypothesis is identified with zeros of N = 4 supersymmetric gauge theory four-point amplitude. The zeros of the ζ(s) function are identified with th complex dimension of the spacetime, or the dimension of the toroidal compactification. A sequence of dimensions are identified in order to map the zeros of the amplitude to the Riemann hypothesis.

We obtain counterparts of the Lindelöf theorem on types entire functions whose growth is determined by Boutroux proximate order. The results are formulated in terms upper density zeros a function and balance its zeros.

These notes constitute a substantially extended version of a talk given in the Fields Institute (Toronto) during the semester \Singularities and Geometry", that culminated by Arnoldfest in celebration of V. I. Arnold's 60th anniversary. We give a survey of diierentresults showing how an upper bound for the numberof isolated zeros for functions satisfying ordinary diierential equations, may be o...

We present combinatorial and analytical results concerning a Sheffer sequence with generating function of the form G(x,z)=Q(z)xQ(−z)1−x, where Q is quadratic polynomial real zeros. By using properties Riordan matrices we address interpretations our polynomials their coefficients. also show that apart from two exceptional zeros, zeros large enough degree in such lie on line x=1/2+it.

A classical result due to Cohn states that a self-inversive polynomial has all its zeros on the unit circle if and only if all the zeros of its derivative lie in the closed unit disk. A more flexible necessary and sufficient condition than that of Cohn’s was given by Chen. However those results do not give any information on the simplicity of zeros of a self-inversive polynomial. This paper mod...