Distribution Of Zeros Research Articles

Entire functions of exponential type represented by pseudo-random and random Taylor series

We study the influence of the multipliers ξ(n) on the angular distribution of zeroes of the Taylor series $${F_\xi }\left( z \right) = \sum\limits_{n \geqslant 0} {\xi \left( n \right)} \frac{{{z^n}}}{{n!}}$$ . We show that the distribution of zeroes of Fξ is governed by certain autocorrelations of the sequence ξ. Using this guiding principle, we consider several examples of random and pseudo-random sequences ξ and, in particular, answer some questions posed by Chen and Littlewood in 1967. As a by-product, we show that if ξ is a stationary random integer-valued sequence, then either it is periodic, or its spectral measure has no gaps in its support. The same conclusion is true if ξ is a complex-valued stationary ergodic sequence that takes values in a uniformly discrete set.

The distribution of zeros of ζ′(s) and gaps between zeros of ζ(s)

Assume the Riemann Hypothesis. We establish a local structure theorem for zeros of the Riemann zeta-function ζ(s) and its derivative ζ′(s). As an application, we prove a stronger form of half of a conjecture of Radziwiłł [18] concerning the global statistics of these zeros. Roughly speaking, we show that on the Riemann Hypothesis, if there occurs a small gap between consecutive zeta zeros, then there is exactly one zero of ζ′(s) lying not only very close to the critical line but also between that pair of zeta zeros. This refines a result of Zhang [22]. Some related results are also shown. For example, we prove a weak form of a conjecture of Soundararajan, and suggest a repulsion phenomena for zeros of ζ′(s).

On multiple orthogonal polynomials for three Meixner measures

Multiple orthogonal polynomials for three discrete Meixner measures with identical exponential decay at infinity are studied. These polynomials are the denominators of the type II Hermite–Pade approximants to some hypergeometric functions. The limit distribution of zeros of such polynomials scaled in a certain way is described in terms of equilibrium logarithmic potentials and in terms of algebraic curves.

Directional Logarithmic Derivative and the Distribution of Zeros of an Entire Function of Bounded L-Index Along the Direction

We establish new criteria of boundedness of the L-index along the direction for entire functions in ℂ n . These criteria are formulated as an estimate for the maximum modulus via the minimum modulus on a circle and also in terms of the restrictions imposed on the distribution of their zeros and on the behavior of the directional logarithmic derivative. In this way, we prove Hypotheses 1 and 2 from [A. I. Bandura and O. B. Skaskiv, Mat. Stud., 43, No. 1, 103–109 (2015)]. The obtained results are also new for the entire functions of bounded index and l-index in ℂ. They improve the well-known results by M. N. Sheremeta, A. D. Kuzyk, and G. H. Fricke.

Distributions of zeros of solutions to first order delay dynamic equations

This paper is concerned with the distributions of zeros of solutions to first order delay dynamic equations on time scales. The results are obtained using iterative sequences.

Distribution of zeros of the S-matrix of chaotic cavities with localized losses and coherent perfect absorption: non-perturbative results

We employ the random matrix theory framework to calculate the density of zeroes of an M-channel scattering matrix describing a chaotic cavity with a single localized absorber embedded in it. Our approach extends beyond the weak-coupling limit of the cavity with the channels and applies for any absorption strength. Importantly it provides an insight for the optimal amount of loss needed to realize a chaotic coherent perfect absorbing trap. Our predictions are tested against simulations for two types of traps: a complex network of resonators and quantum graphs.

The Properties of Differential-Difference Polynomials

The main aim of the paper is to improve some classical results on the distribution of zeros for differential polynomials and differential-difference polynomials. We present some results on the distribution of zeros of [f(z) n f(z + c)](k) − α(z) and [f(z) n (f(z + c) − f(z))](k) − α(z) and give some examples to show that the results are best possible in a certain sense.

On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials

This survey considers multiple orthogonal polynomials with respect to Nikishin systems generated by a pair of measures with unbounded supports (, ) and with discrete. A Nikishin-type equilibrium problem in the presence of an external field acting on and a constraint on is stated and solved. The solution is used for deriving the contracted zero distribution of the associated multiple orthogonal polynomials. Bibliography: 56 titles.

Zero distribution of random sparse polynomials

We study asymptotic zero distribution of random Laurent polynomials whose support are contained in dilates of a fixed integral polytope $P$ as their degree grow. We consider a large class of probability distributions including the ones induced from i.i.d. random coefficients whose distribution law has bounded density with logarithmically decaying tails as well as moderate measures defined over the projectivized space of Laurent polynomials. We obtain a quantitative localized version of Bernstein-Kouchnirenko Theorem.

Tail bounds for counts of zeros and eigenvalues, and an application to ratios

Let t be random and uniformly distributed in the interval [T,2T] , and consider the quantity N(t+1/\log\:T) - N(t) , a count of zeros of the Riemann zeta function in a box of height 1/\mathrm {log}\: T . Conditioned on the Riemann hypothesis, we show that the probability this count is greater than x decays at least as quickly as e^{-Cx\:\mathrm {log}\: x} , uniformly in T . We also prove a similar results for the logarithmic derivative of the zeta function, and likewise analogous results for the eigenvalues of a random unitary matrix. We use results of this sort to show on the Riemann hypothesis that the averages [ \frac{1}{T} \int_T^{2T} \Bigg| \frac{\zeta\big(\frac{1}{2} + \frac{\alpha}{\log T} + it\big)}{\zeta\big(\frac{1}{2}+ \frac{\beta}{\log T} + it\big)}\Bigg|^m,dt ] remain bounded as T\rightarrow\infty , for \alpha, \beta complex numbers with \beta\neq 0 . Moreover we show rigorously that the local distribution of zeros asymptotically controls ratio averages like the above; that is, the GUE Conjecture implies a (first-order) ratio conjecture.

Zeros of Sections of Power Series: Deterministic and Random

We present a streamlined proof (and some refinements) of a characterization (due to F. Carlson and G. Bourion, and also to P. Erdős and H. Fried) of the so-called Szegő power series. This characterization is then applied to readily obtain some (more) recent known results and some new results on the asymptotic distribution of zeros of sections of random power series, extricating quite naturally the deterministic ingredients. Finally, we study the possible limits of the zero counting probabilities of a power series.

Discontinuity in the Asymptotic Behavior of Planar Orthogonal Polynomials Under a Perturbation of the Gaussian Weight

We consider the orthogonal polynomials, $\{P_n(z)\}_{n=0,1,\cdots}$, with respect to the measure $$|z-a|^{2c} e^{-N|z|^2}dA(z)$$ supported over the whole complex plane, where $a>0$, $N>0$ and $c>-1$. We look at the scaling limit where $n$ and $N$ tend to infinity while keeping their ratio, $n/N$, fixed. The support of the limiting zero distribution is given in terms of certain "limiting potential-theoretic skeleton" of the unit disk. We show that, as we vary $c$, both the skeleton and the asymptotic distribution of the zeros behave discontinuously at $c=0$. The smooth interpolation of the discontinuity is obtained by the further scaling of $c=e^{-\eta N}$ in terms of the parameter $\eta\in[0,\infty).$

Phase space distribution of Riemann zeros

We present the partition function of a most generic U(N) single plaquette model in terms of representations of a unitary group. Extremising the partition function in a large N limit, we obtain a relation between eigenvalues of unitary matrices and the number of boxes in the most dominant Young tableaux distribution. Since the eigenvalues of unitary matrices behave like coordinates of free fermions, whereas the number of boxes in a row is like conjugate momenta of the same, a relation between them allows us to provide a phase space distribution for different phases of the unitary model under consideration. This proves a universal feature that all the phases of a generic unitary matrix model can be described in terms of topology of free fermi phase space distribution. Finally, using this result and analytic properties of resolvent that satisfy the Dyson-Schwinger equation, we present a phase space distribution of unfolded zeros of the Riemann zeta function.

On the distribution of the zeros of the Hermite–Padé polynomials for a quadruple of functions

On the distribution of the zeros of the Hermite–Padé polynomials for a quadruple of functions

Partition and generating function zeros in adsorbing self-avoiding walks

The Lee–Yang theory of adsorbing self-avoiding walks is presented. It is shown that Lee–Yang zeros of the generating function of this model asymptotically accumulate uniformly on a circle in the complex plane, and that Fisher zeros of the partition function distribute in the complex plane such that a positive fraction are located in annular regions centred at the origin. These results are examined in a numerical study of adsorbing self-avoiding walks in the square and cubic lattices. The numerical data are consistent with the rigorous results; for example, Lee–Yang zeros are found to accumulate on a circle in the complex plane and a positive fraction of partition function zeros appear to accumulate on a critical circle. The radial and angular distributions of partition function zeros are also examined and it is found to be consistent with the rigorous results.

The Euler product for the Riemann zeta-function in the critical strip

In this paper we study a pointwise asymptotic behavior of the partial Euler product for the Riemann zeta-function on the right half of the critical strip. We discuss relations among the behavior of the partial Euler product, the distribution of the prime numbers and the distribution of nontrivial zeros of the Riemann zeta-function.

Nevanlinna theory of the Wilson divided-difference operator

Sitting at the top level of the Askey-scheme, Wilson polynomials are regarded as the most general hypergeometric orthogonal polynomials. Instead of a differential equation, they satisfy a second order Sturm-Liouville type difference equation in terms of the Wilson divided-difference operator. This suggests that in order to better understand the distinctive properties of Wilson polynomials and related topics, one should use a function theory that is more natural with respect to the Wilson operator. Inspired by the recent work of Halburd and Korhonen, we establish a full-fledged Nevanlinna theory of the Wilson operator for meromorphic functions of finite order. In particular, we prove a Wilson analogue of the lemma on logarithmic derivatives, which helps us to derive Wilson operator versions of Nevanlinna's Second Fundamental Theorem, some defect relations and Picard's Theorem. These allow us to gain new insights on the distributions of zeros and poles of functions related to the Wilson operator, which is different from the classical viewpoint. We have also obtained a relevant five-value theorem and Clunie type theorem as applications of our theory, as well as a pointwise estimate of the logarithmic Wilson difference, which yields new estimates to the growth of meromorphic solutions to some Wilson difference equations and Wilson interpolation equations.

Distribution of zeta zeros and the oscillation of the error term of the prime number theorem

An 84-year-old classical result of Ingham states that a rather general zero-free region of the Riemann zeta function implies an upper bound for the absolute value of the remainder term of the prime number theorem. In 1950 Tur´an proved a partial conversion of the mentioned theorem of Ingham. Later the author proved sharper forms of both Ingham’s theorem and its conversion by Tur´an. The present work shows a very general theorem which describes the average and the maximal order of the error terms by a relatively simple function of the distribution of the zeta zeros. It is proved that the maximal term in the explicit formula of the remainder term coincides with high accuracy with the average and maximal order of the error term.

Calculation of the Riemann Zeta-function on a Relativistic Computer

The problem of calculating the sum of a divergent series for the Riemann ζ-function of a complex argument is considered in the paper, using the effects of the general theory of relativity. The parameters of the reference frame metric in which the calculation is performed are determined and solutions of the equations of motion of the material point realizing the calculation are found. The work lies at the junction of the direction known as Beyond Turing, considering the application of the so-called relativistic supercomputers for solving non-computable problems and a direction devoted to the study of non-trivial zeros of the Riemann ζ-function. The formulation of the Riemann hypothesis concerning the distribution of nontrivial zeros of the ζ-function from the point of view of their computability on a computer is given. In view of the importance of the latter issue for studying the distribution of prime numbers, the results of the work may be of interest to specialists in the field of information security.

The Proof of the Riemann Hypothesis on a Relativistic Turing Machine

In this article, the proof of the Riemann hypothesis is considered using the calculation of the Riemann ζ-function on a relativistic computer. The work lies at the junction of the direction known as "Beyond Turing", considering the application of the so-called "relativistic supercomputers" for solving non-computable problems and a direction devoted to the study of non-trivial zeros of the Riemann ζ-function. Considerations are given in favor of the validity of the Riemann hypothesis with respect to the distribution of non-trivial zeros of the ζ-function.