This subject started with H. L. Montgomery’s famous paper [12], where he introduced a method to study the vertical distribution of the zeros of the Riemann zeta-function. Assuming the Riemann Hypothesis, he calculated the pair correlation of the zeros and from it deduced an estimate for the size of small gaps between the zeros and for the percentage of simple zeros. This paper opened a whole ne...

The Yang-Lee zeros of the Q-state Potts model are investigated in 1, 2 and 3 dimensions. Analytical results derived from the transfer matrix for the one-dimensional model reveal a systematic behavior of the locus of zeros as a function of Q. For 1 < Q < 2 the zeros in the complex x = exp(βHq) plane lie inside the unit circle, while for Q > 2 they lie outside the unit circle for finite temperatu...

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...

The Standard Model amplitudes for processes where one or more gauge bosons are emitted exhibit zeros in the angular distributions. The theoretical and experimental aspects of these radiation amplitude zeros are reviewed and some recent results are discussed. In particular, the zeros of the WZγ and WZZ production amplitudes are analyzed. It is briefly explained how radiation zeros can be used to...

The Yang-Lee, Fisher and Potts zeros of the one-dimensional Q-state Potts model are studied using the theory of dynamical systems. An exact recurrence relation for the partition function is derived. It is shown that zeros of the partition function may be associated with neutral fixed points of the recurrence relation. Further, a general equation for zeros of the partition function is found and ...

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 a 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 “Man findet nun in der That etwa so viel reelle Wurzeln innerhalb dieser Grenzen, und es ist sehr wahrscheinlich, dass ...

It is well-known and easy to see that the zeros of both the associated polynomial and the derivative of an orthogonal polynomial pn(x) interlace with the zeros of pn(x) itself. The natural question of how these zeros interlace is under discussion. We give a sufficient condition for the mutual location of k-th, 1 ≤ k ≤ n − 1, zeros of the associated polynomial and the derivative of an orthogonal...

which is valid for any complex s, it follows that ζ(s) has zeros at s = −2,−4, . . . . These zeros are traditionally called the “trivial” zeros of ζ(s), to distinguish them from the complex zeros of ζ(s), of which the smallest ones (in absolute value) are 12 ± 14.134725 . . . i. It is well-known that all complex zeros of ζ(s) lie in the so-called “critical strip” 0 < σ = Re s < 1, and the Riema...

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...