On universal relatives of the Riemann zeta-function

The Riemann zeta-function ζ has the following well-known properties (M) It is meromorphic in ℂ with a simple pole at z = 1 with residue 1.     

A generalized square of the zeta function. Spectral decompositions

The spectral decomposition for the square of the classical Riemann zeta function ζ²(s) is generalized to the case of the product of two such functions ζ(s₁) · ζ(s₂) of different arguments. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 17–44.     

Projective version of Malcolmson’s criterion

It is proven that the extension of Malcolmson’s methods to reppresent all elements ofR _ζ is not purely straightforward. A detailed proof of such extension is here furnished.

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Sunto Si prova che l’estensione dei metodi di Malcolmson per la rappresentazione degli elementi diR _ζ a tutte le applicazioni tra moduli proiettivi indotti non è puramente meccanica. è fornita una dimostrazione dettagliata di detta estensione.

     

Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ^-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq. Work supported in part by an operating grant from NSERC (Canada).     

Generalizations of a discrete universality theorem for Hurwitz zeta-functions

It is well known that the Hurwitz zeta-function ζ(s, α) with transcendental or rational parameter α is universal in the sense that the shifts ζ(s + iτ ), t ? \mathbbR \tau \in \mathbb{R} (continuous case), and ζ(s + imh), m ? \mathbbN è{ 0 } m \in \mathbb{N} \cup \left\{ 0 \right\} , with fixed h > 0 (discrete case) approximate any analytic function. In the paper, the discrete universality is extended for some classes of the functions F(ζ(s, α)).     

On limit distribution of the Hurwitz zeta-function

The distribution of the vector (|ζ(s, α)|; ζ(s, α)), where ζ(s, α) is the Hurwitz zeta-function with transcendental parameter α, is considered and a probabilistic limit theorem is obtained. Also, the dependence between |ζ(s, α)| and ζ(s, α) in terms of m-characteristic transforms is discussed.     

On the distribution of values of the derivative of the Riemann zeta function at its zeros. I

Let ζ′(s) be the derivative of the Riemann zeta function ζ(s). A study on the value distribution of ζ′(s) at the non-trivial zeros ρ of ζ(s) is presented. In particular, for a fixed positive number X, an asymptotic formula and a non-trivial upper bound for the sum Σ_{0<Im ρ≤T} ζ′(ρ)X ^ρ as T → ∞ are given. We clarify the dependence on the arithmetic nature of X.     

Shadowing and the Plasma-Sheath Transition Layer

Summary. This paper gives a rigorous derivation of the system μ ² d ² w ₀ \over dζ ² =w ² ₀ +ζ , w ₀ (ζ ₀ )=-(-ζ ₀ ) ^1/2 , dw ₀ \over dζ (ζ ₀ )= 1\over 2 (-ζ ₀ ) ^-1/2 , ζ ₀ <0 , governing the electric potential in the transition layer joining quasi-neutral plasma to space charge sheath in a weakly ionized plasma. The parameter μ represents the tolerance for ``shadowing,' i.e., it measures the ``distance' between the true solution of the full Euler-Poisson system and the solution of the reduced order limit equation for w ₀ (ζ) . Received February 22, 2001; accepted September 21, 2001 Online publication November 30, 2001     

Nesterenko’s criterion when the small linear forms oscillate

In this paper we generalize Nesterenko’s criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the vector space spanned over the rationals by a family of real numbers and a measure of simultaneous approximation to these numbers (namely, an upper bound for the irrationality exponent if 1 and only one other number are involved). As an application, we prove an explicit measure of simultaneous approximation to ζ(5), ζ(7), ζ(9), and ζ(11), using Zudilin’s proof that at least one of these numbers is irrational.     

Conjugation properties in incidence algebras

Incidence algebras can be regarded as a generalization of full matrix algebras. We present some conjugation properties for incidence functions. The list of results is as follows: a criterion for a convexdiagonal function f to be conjugated to the diagonal function fe; conditions under which the conjugacy f ∼ Ce + ζ _⋖-holds (the function Ce + ζ _⋖-may be thought of as an analog for a Jordan box from matrix theory); a proof of the conjugation of two functions ζ _< and gz _⋖-for partially ordered sets that satisfy the conditions mentioned above; an example of a partially ordered set for which the conjugacy ζ _< ∼ ζ _⋖-does not hold. These results involve conjugation criteria for convex-diagonal functions of some partially ordered sets. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 111–123, 2003.     

ζ μ -sets in generalized topological spaces

We define ζ _μ -sets, (ζ,μ)-closed sets and generalized ζ _μ -sets in a generalized topological space and investigate properties of several low separation axioms of generalized topologies constructed by the families of these sets. Characterizations of some properties of (ζ,μ)-R ₀ and (ζ,μ)-R ₁ generalized topological spaces will be given.     

Cyclotomic and simplicial matroids

We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid μ_n represented by then ^th roots of unity 1, ζ, ζ², …, ζ^n-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among then ^th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of μ_n in the case thatn has two prime factors. First author supported by NSF Postdoctoral Fellowship. Second author supported by NSF grant DMS-0245379.     

Universality of random matrices and local relaxation flow

Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we prove that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N ^−ζ for some ζ>0. The proof is based on an estimate of the entropy flow of the Dyson Brownian motion w.r.t. a “pseudo equilibrium measure”. As an application of this estimate, we prove that the eigenvalue spacing statistics in the bulk of the spectrum for N×N symmetric Wigner ensemble is the same as that of the Gaussian Orthogonal Ensemble (GOE) in the limit N→∞. The assumptions on the probability distribution of the matrix elements of the Wigner ensemble are a subexponential decay and some minor restriction on the support.     

A characterization of convexity and central symmetry for planar polygonal sets

LetS be a compact set inR ² with nonempty interior,L(u,k) be a line 〈u, x〉 =k, and ζ _u (k) be the linear Lebesgue measure ofS∩L(u,k). It is well known that for a convexS, ζ _u (k) is unimodal, that is, as a function ofk, it is first non-decreasing and then nonincreasing for everyu∈R ². Further, ifS is centrally symmetric with respect toM, ζ _u (k) achieves maximum whenL(u, k) passes throughM. Converse propositions are considered in this paper for polygonalS with Jordan boundary. It is shown that unimodality alone does not suffice for convexity. However, if ζ _u (k) achieves maximum wheneverL(u, k) passes through some fixed pointM then unimodality yields convexity as well as central symmetry. It is also shown that continuity of ζ _u (k) in the interior of its support implies convexity ofS. This last result, however, is false for non-polygonal sets. Research supported by National Science Foundation Grant GP-28154.     

Branching structure of uniform recursive trees

The branching structure of uniform recursive trees is investigated in this paper. Using the method of sums for a sequence of independent random variables, the distribution law of ηn, the number of branches of the uniform recursive tree of size n are given first. It is shown that the strong law of large numbers, the central limit theorem and the law of iterated logarithm for ηn follow easily from this method. Next it is shown that ηn and ξn, the depth of vertex n, have the same distribution, and the distribution law of ζn,m, the number of branches of size m, is also given, whose asymptotic distribution is the Poisson distribution with parameter λ= 1/m. In addition, the joint distribution and the asymptotic joint distribution of the numbers of various branches are given. Finally, it is proved that the size of the biggest branch tends to infinity almost sure as n→∞.     

Cohomological growth rates and Kazhdan-Lusztig polynomials

In a previous work, the authors established various bounds for the dimensions of degree n cohomology and Ext-groups, for irreducible modules of semisimple algebraic groups G (in positive characteristic p) and (Lusztig) quantum groups U _ζ (at roots of unity ζ). These bounds depend only on the root system, and not on the characteristic p or the size of the root of unity ζ. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system ϕ. For quantum groups U _ζ with a fixed ϕ, we show the sequence {max _{L irred} dim H ⁿ (U _ζ , L)} _n has polynomial growth independent of ζ. In fact, we provide upper and lower bounds for the polynomial growth rate. Applications of these and related results for are given to Kazhdan-Lusztig polynomials. Polynomial growth in the algebraic group case remains an open question, though it is proved that {log max _{L irred} dim H ⁿ (G,L)} has polynomial growth ≤ 3 for any fixed prime p (and ≤ 4 if p is allowed to vary with n). We indicate the relevance of these issues to (additional structure for) the constants proposed in the theory of higher cohomology groups for finite simple groups with irreducible coefficients by Guralnick, Kantor, Kassabov and Lubotzky [13].     

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

Let θ(ζ) be a Schur operator function, i.e., it is defined and holomorphic on the unit disk := C : 1 {\mathbb {D} := \{\zeta \in \mathbb {C} : \vert\zeta\vert < 1 \}} and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and *-outer Schur operator functions j(z){\varphi(\zeta)} and ψ(ζ) which describe respectively the deviations of the function θ(ζ) from inner and *-inner operator functions are studied. If j(z) 1 0{\varphi(\zeta)\neq 0} , then it means that in the scattering system for which θ(ζ) is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system (Sect. 6). The function ψ(ζ) has the analogous property for the dual system. For this reason these functions are called the defect functions of the function θ(ζ). The explicit form of the defect functions j(z){\varphi(\zeta)} and ψ(ζ) is obtained and the analytic connection of these functions with the function θ(ζ) is described (Sects. 3, 5). The operator functions (l j(z)q(z)){\left(\begin{array}{l} \varphi(\zeta)\\ \theta(\zeta)\end{array}\right)} and (ψ(ζ), θ(ζ)) are Schur functions as well (Sect. 3). It is important that there exists the unique contractive measurable operator function χ(t), t ? ?\mathbb D{t\in\partial\mathbb {D}} , such that the operator function (l c(t)    j(t)y(t)    q(t) ){\left(\begin{array}{l} \chi(t)\quad \varphi(t)\\ \psi(t)\quad \theta(t) \end{array}\right)} , t ? ?\mathbb D,{t\in\partial\mathbb {D},} is also contractive (Part II, Sect. 12). The second part of the paper is devoted to studying the properties of the function χ(t). Specifically, it is shown that the function χ(t) is the scattering suboperator through the internal channels of the scattering system for which θ(ζ) is the transfer function (Part II, Sect. 12).     

Extreme Bloch Functions and Summation Methods for Bloch Functions

In this paper we construct Bloch functions F for which the set {z | e sup _{|ζ| < 1} |F'(ζ)| ( 1 - |ζ| ² ) = |F'(z)| ( 1 - |z| ² )} is an analytic Jordan curve tangential to the unit disk in some points. It is proved that, using such functions, we can derive analogs to the Taylor expansion for Bloch functions in cases where the Taylor expansion does not converge. October 15, 1997. Date revised: March 12, 1998. Date accepted: June 18, 1998.     

On Critical Values of Polynomials with Real Critical Points

Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that |f(ζ)/ζ|≤2/3, and that this inequality can be taken to be strict unless f is of the form f(z)=z+cz ³.