Distribution Of Zeros Research Articles

On equilibrium problems related to the distribution of zeros of the Hermite-Padé polynomials

We study two potential-theory equilibrium problems that arise naturally in the theory of the limit distribution of zeros of the Hermite–Pade polynomials. We analyze the relationship between these problems and prove that the equilibrium measure for one of the problems is the balayage of the equilibrium measure for the other problem.

Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomials

We study the asymptotic behavior of the zeros of a family of a certain class of hypergeometric polynomials $$\small {}_{A}\text {F}_{B}\left[ \begin{array}{c} -n,a_2,\ldots ,a_A \\ b_1,b_2,\ldots ,b_B \end{array} ; \begin{array}{cc} z \end{array}\right] $$ , using the associated hypergeometric differential equation, as the parameters go to infinity. The curve configuration on which the zeros cluster is characterized as level curves associated with integrals on an algebraic curve. The algebraic curve is the hypergeometrc differential equation, using a similar approach to the method used in Borcea et al. (Publ Res Inst Math Sci 45(2):525–568, 2009). In a specific degenerate case, we make a conjecture that generalizes work in Boggs and Duren (Comput Methods Funct Theory 1(1):275–287, 2001), Driver and Duren (Algorithms 21(1–4):147–156, 1999), and Duren and Guillou (J Approx Theory 111(2):329–343, 2001), and present experimental evidence to substantiate it.

Asymptotic Behavior and Zero Distribution of Polynomials Orthogonal with Respect to Bessel Functions

We consider polynomials $$P_n$$ orthogonal with respect to the weight $$J_{\nu }$$ on $$[0,\infty )$$ , where $$J_{\nu }$$ is the Bessel function of order $$\nu $$ . Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros of $$P_n$$ are complex and accumulate as $$n \rightarrow \infty $$ near the vertical line $${{\mathrm{{\text {Re}}\,}}}z = \frac{\nu \pi }{2}$$ . We prove this fact for the case $$0 \le \nu \le 1/2$$ from strong asymptotic formulas that we derive for the polynomials $$P_n$$ in the complex plane. Our main tool is the Riemann–Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift–Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for $$\nu \le 1/2$$ .

Fine structure in the large [formula omitted] limit of the non-Hermitian Penner matrix model

In this paper we apply results on the asymptotic zero distribution of the Laguerre polynomials to discuss generalizations of the standard large n limit in the non-Hermitian Penner matrix model. In these generalizations gnn→t, but the product gnn is not necessarily fixed to the value of the ’t Hooft coupling t. If t>1 and the limit l=limn→∞|sin(π/gn)|1/n exists, then the large n limit is well-defined but depends both on t and on l. This result implies that for t>1 the standard large n limit with gnn=t fixed is not well-defined. The parameter l determines a fine structure of the asymptotic eigenvalue support: for l≠0 the support consists of an interval on the real axis with charge fraction Q=1−1/t and an l-dependent oval around the origin with charge fraction 1/t. For l=1 these two components meet, and for l=0 the oval collapses to the origin. We also calculate the total electrostatic energy E, which turns out to be independent of l, and the free energy F=E−Qlnl, which does depend on the fine structure parameter l. The existence of large n asymptotic expansions of F beyond the planar limit as well as the double-scaling limit are also discussed.

Thermodynamics of the bosonic randomized Riemann gas

The partition function of a bosonic Riemann gas is given by the Riemann zeta function. We assume that the Hamiltonian of this gas at a given temperature has a random variable ω with a given probability distribution over an ensemble of Hamiltonians. We study the average free energy density and average mean energy density of this arithmetic gas in the complex β-plane. Assuming that the ensemble is made by an enumerable infinite set of copies, there is a critical temperature where the average free energy density diverges due to the pole of the Riemann zeta function. Considering an ensemble of non-enumerable set of copies, the average free energy density is non-singular for all temperatures, but acquires complex values in the critical region. Next, we study the mean energy density of the system which depends strongly on the distribution of the non-trivial zeros of the Riemann zeta function. Using a regularization procedure we prove that the this quantity is continuous and bounded for finite temperatures.

Exact Maximal Convergence in Capacity and Zero Distribution of Rational Approximants

Let E be a compact, connected set in \(\mathbb {C}\) with regular, connected complement, and let f be meromorphic on E with maximal Green domain \(E_{\rho (f)}\) of meromorphy, \(\rho (f) < \infty \). We investigate rational approximations \(r_{n,m_n}\) of f on E with numerator degree \(\le n\) and denominator degree \(\le m_n\) where \(m_n = o(n)\) as \(n \rightarrow \infty \) and $$\begin{aligned} \underset{n \rightarrow \infty }{\lim \sup } \, \Vert f - r_{n,m_n}\Vert ^{1/n}_{\partial E} \le \frac{1}{\rho (f)}. \end{aligned}$$ If \(\{r_{n,m_n}\}_{n \in \Lambda }\) is maximally convergent in capacity to f on \(E_{\rho (f)}\) for some \(\Lambda \subset \mathbb {N}\), then this subsequence converges exactly to f on \(E_{\rho (f)}\) if f has a branch point on the boundary of \(E_{\rho (f)}\). Moreover, the exact convergence in capacity implies that the normalized zero distributions of \(r_{n,m_n}\) converge weakly to the equilibrium measure of \(\overline{E}_{\rho (f)}\) as \(n \rightarrow \infty \), at least for a subsequence of \(\mathbb {N}\).

Certain Polynomials with Weighted Sums

AbstractIn this note, we provide some examples of polynomials z n −p(z), where , and , a k ≥ 0 for eachk such that p(z) has all its zeros on |z|=c 1, and wedenote C(r) by the circle of radius r with center the ori-gin. It follows from Enestrm-Kakeya theorem [1] (see p.136 of [1] for the statement and its proof) to(1)where , a k ≥ 0 for each k that all zeros of (1) do not lie outside C(1). To what extent, are there pol-ynomialswhere , a k ≥ 0 for each k with all its zeros on |z|=c<1 such that z n −p(z) has all its zeros on two cir-cles |z| = 1 and |z|=d< 1? Whether or not certain pol-ynomials have all their zeros on some circles is one ofthe most fundamental questions in the theory of distri-bution of polynomial zeros. Kim [2] studied polynomialsof type (1),whose all zeros except for z= 1 lie on C(1/p). For con-venience, we call these polynomials C(1/p)-polynomials, and their weighted sums, respectively.Kim

Lee-Yang zero distribution of high temperature QCD and the Roberge-Weiss phase transition

Canonical partition functions and Lee-Yang zeros of QCD at finite density and high temperature are studied. Recent lattice simulations have confirmed that the free energy of QCD is a quartic function of quark chemical potential at temperature slightly above pseudo-critical temperature $T_c$, as in the case with a gas of free massless fermions. We present analytic derivation of the canonical partition functions and Lee-Yang zeros for this type of free energy using the saddle point approximation. We also perform lattice QCD simulation in a canonical approach using the fugacity expansion of the fermion determinant, and carefully examine its reliability. By comparing the analytic and numerical results, we conclude that the canonical partition functions follow the Gaussian distribution of the baryon number, and the accumulation of Lee-Yang zeros of these canonical partition functions exhibit the first-order Roberge-Weiss phase transition. We discuss the validity and applicable range of the result and its implications both for theoretical and experimental studies.

Asymptotic properties of zeta functions over finite fields

In this paper we study asymptotic properties of families of zeta and L-functions over finite fields. We do it in the context of three main problems: the basic inequality, the Brauer–Siegel type results and the results on distribution of zeroes. We generalize to this abstract setting the results of Tsfasman, Vlăduţ and Lachaud, who studied similar problems for curves and (in some cases) for varieties over finite fields. In the classical case of zeta functions of curves we extend a result of Ihara on the limit behaviour of the Euler–Kronecker constant. Our results also apply to L-functions of elliptic surfaces over finite fields, where we approach the Brauer–Siegel type conjectures recently made by Kunyavskii, Tsfasman and Hindry.

English

Let $S_k$ be the space of holomorphic cusp forms of weight $k$ with respect to $SL_2(\mathbb{Z})$. Let $f \in S_k$ be a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper we consider the distribution of zeros of $L_f(s)$ in the strip $\sigma \leq \Re s \leq 1$ for fixed $\sigma>1/2$ with respect to the imaginary part. We study estimates of \[ N_f(\sigma,T) = #\{\rho\in\mathbb{C} \mid L_f(\rho)=0, leq \Re\rho \leq 1, 0 \leq \Im\rho \leq T} \] for $1/2 \leq \sigma \leq1$ and large $T>0$. Using the methods of Karatsuba and Voronin we shall give another proof for Ivic's method.

The Polynomials of Mahler and Roots of Unity

Kurt Mahler in 1982 studied a special sequence of very sparse (0,1)-polynomials which have only powers of 2 as exponents. In this paper we study divisibility properties of these polynomials by certain cyclotomic polynomials and prove an explicit version of a result that was given only implicitly by Mahler. We also consider the distribution of real and complex noncyclotomic zeros, improving some of Mahler’s results. Then we show that the derivatives of the polynomials of Mahler have all their zeros inside the unit circle. We conclude this paper with some further remarks and open questions.

The distribution of zeros of all solutions of first order neutral differential equations

This paper is concerned with the distribution of zeros of all solutions of the first-order neutral differential equationx(t)+p(t)x(t-τ)′+Q(t)x(t-σ)=0,t⩾t0,wherep∈C[t0,∞),[0,∞),Q∈C[t0,∞),(0,∞)andτ,σ∈R+.New estimations for the distance between adjacent zeros of this neutral equation are obtained via comparison with a corresponding differential inequality. These results extend some known results from the non-neutral to the neutral case and improve other published results as well.

The Distribution of Zeros of Quadratic Forms over Finite Fields

Let $m$ be a positive integer with $m &amp;lt; p/2$ and $p$ is a prime. Let $\mathbb{F}_q$ be the finite field in $q = p^f$ elements, $Q({\mathbf{x}})$ be a nonsinqular quadratic form over $\mathbb{F}_q$ with $q$ odd, $V$ be the set of points in $\mathbb{F}_q^n$ satisfying the equation $Q({\mathbf{x}}) = 0$ in which the variables are restricted to a box of points of the type\[\mathcal{B}(m) = \left\{ {{\mathbf{x}} \in \mathbb{F}_q^n \left| {x_i = \sum\limits_{j = 1}^f {x_{ij} \xi _j } ,\;\left| {x_{ij} } \right| &amp;lt; m,\;1 \leqslant i \leqslant n,\;1 \leqslant j \leqslant f} \right.} \right\},\]where $\xi _1 , \ldots ,\xi _f$ is a basis for $\mathbb{F}_q$ over $\mathbb{F}_p$ and $n &amp;gt; 2$ even. Set $\Delta = \det Q$ such that $\chi \left( {( - 1)^{n/2} \Delta } \right) = 1.$ We shall motivate work of (Cochrane, 1986) to obtain lower bounds on $m,$ size of the box $\mathcal{B},$ so that $\mathcal{B} \cap V$ is nonempty. For this we show that the box $\mathcal{B}(m)$ contains a zero of $Q({\mathbf{x}})$ provided that $m \geqslant p^{1/2}.$ We also show that the box $\mathcal{B}(m)$ contains $n$ linearly independent zeros of $Q({\mathbf{x}})$ provided that $m \geqslant 2^{n/2} p^{1/2} .$

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let $$G$$ be a reductive group over a number field $$F$$ which admits discrete series representations at infinity. Let $$^{L}G=\widehat{G} \rtimes \mathrm{Gal}(\bar{F}/F)$$ be the associated $$L$$ -group and $$r:{}^L G\rightarrow \mathrm{GL}(d,\mathbb {C})$$ a continuous homomorphism which is irreducible and does not factor through $$\mathrm{Gal}(\bar{F}/F)$$ . The families under consideration consist of discrete automorphic representations of $$G(\mathbb {A}_F)$$ of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato–Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321–331, 1987) and Serre (J Am Math Soc 10(1):75–102, 1997). As an application we study the distribution of the low-lying zeros of the associated family of $$L$$ -functions $$L(s,\pi ,r)$$ , assuming from the principle of functoriality that these $$L$$ -functions are automorphic. We find that the distribution of the $$1$$ -level densities coincides with the distribution of the $$1$$ -level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz–Sarnak heuristics. We provide a criterion based on the Frobenius–Schur indicator to determine this symmetry type. If $$r$$ is not isomorphic to its dual $$r^\vee $$ then the symmetry type is unitary. Otherwise there is a bilinear form on $$\mathbb {C}^d$$ which realizes the isomorphism between $$r$$ and $$r^\vee $$ . If the bilinear form is symmetric (resp. alternating) then $$r$$ is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

On the Number of Zeros of Linear Combinations of Independent Characteristic Polynomials of Random Unitary Matrices

We show that almost all the zeros of any finite linear combination of inde- pendent characteristic polynomials of random unitary matrices lie on the unit circle. This result is the random matrix analogue of an earlier result by Bombieri and Hejhal on the distribution of zeros of linear combinations of L-functions, thus providing further evidence for the conjectured links between the value distribution of the characteristic polynomial of random unitary matrices and the value distribution of L-functions on the critical line.

Local spectral equidistribution for degree two Siegel modular forms in level and weight aspects

We prove an equidistribution result for the Satake parameters of the local representations attached to Siegel cusp forms of degree 2 of increasing level and weight, counted with a certain arithmetic weight. We then apply this to compute the symmetry type of a similarly weighted distribution of the low-lying zeros of L-functions attached to these cusp forms.

Complete monotonicity and zeros of sums of squared Baskakov functions

We prove complete monotonicity of sums of squares of generalized Baskakov basis functions by deriving the corresponding results for hypergeometric functions. Moreover, in the central Baskakov case we study the distribution of the complex zeros for large values of a parameter. We finally discuss the extension of some results for sums of higher powers.

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

We investigate the asymptotic behaviour of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree d + 1. The polynomials that we investigate are multiple orthogonal with respect to a system of d analytic weights defined on a symmetric (d + 1)-star centred at the origin. In the first part we analyse in detail a vector equilibrium problem involving a system of d interacting measures (μ1, …, μd) supported on star-like sets in the plane. We show that in the subcritical regime, the first component of the solution to this problem is the asymptotic zero distribution of the multiple orthogonal polynomials. It also characterizes the domain where the eigenvalues in the normal matrix model accumulate, in the sense that the Schwarz function associated with the boundary of this domain can be expressed explicitly in terms of . The second part of the paper is devoted to the asymptotic analysis of the multiple orthogonal polynomials. The asymptotic results are obtained again in the subcritical regime, and they follow from the Deift/Zhou steepest descent analysis of a Riemann–Hilbert problem of size (d + 1) × (d + 1). The vector equilibrium problem and the Riemann–Hilbert problem that we investigate are generalizations of those studied recently by Bleher–Kuijlaars in the case d = 2.

Fourier Transforms of Positive Definite Kernels and the Riemann $$\xi $$ ξ -Function

The purpose of this paper is to investigate the distribution of zeros of entire functions which can be represented as the Fourier transforms of certain admissible kernels. The principal results bring to light the intimate connection between the Bochner–Khinchin–Mathias theory of positive definite kernels and the generalized real Laguerre inequalities. The concavity and convexity properties of the Jacobi theta function play a prominent role throughout this work. The paper concludes with several questions and open problems.

Распределение нулей обобщенных полиномов Эрмита

Распределение нулей обобщенных полиномов Эрмита