Sharpenings of Li's Criterion for the Riemann Hypothesis

Exact and asymptotic formulae are displayed for the coefficients λ _n used in Li's criterion for the Riemann Hypothesis. For n → ∞ we obtain that if (and only if) the Hypothesis is true, λ _n ∼ n(A log n + B) (with A > 0 and B explicitly given, also for the case of more general zeta or L-functions); whereas in the opposite case, λ _n has a non-tempered oscillatory form. ^★Institut de Mathématiques de Jussieu-Chevaleret (CNRS UMR 7586), Université Paris 7, F-75251 Paris CEDEX 05, France.     

The phase of the Riemann zeta function

We, offer an alternative interpretation of the Riemann zeta functionζ(s) as a scattering amplitude and its nontrivial zeros as the resonances in the scattering amplitude. We also look at several different facets of the phase of theζ function. For example, we show that the smooth part of theζ function along the line of the zeros is related to the quantum density of states of an inverted oscillator. On the other hand, for ℜs>1/2, we show that the memory of the zeros fades only gradually through a Lorentzian smoothing of the delta functions. The corresponding trace formula for ℜs≫1 is shown to be of the same form as generated by a one-dimensional harmonic oscillator in one direction along with an inverted oscillator in the transverse direction. Quite remarkably for this simple model, the Gutzwiller trace formula can be obtained analytically and is found to agree with the quantum result.     

Hierarchy of the Selberg Zeta Functions

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.Mathematics Subject Classifications (2000). Primary 11M36, Secondary 33B15     

Inverse Scattering, the Coupling Constant Spectrum, and the Riemann Hypothesis

It is well known that the s-wave Jost function for a potential, V, is an entire function of with an infinite number of zeros extending to infinity. For a repulsive V, and at zero energy, these zeros of the coupling constant, , will all be real and negative, _n (0)<0. By rescaling , such that _n <–1/4, and changing variables to s, with =s(s–1), it follows that as a function of s the Jost function has only zeros on the line s _n =1/2+i _n . Thus, finding a repulsive V whose coupling constant spectrum coincides with the Riemann zeros will establish the Riemann hypothesis, but this will be a very difficult and unguided search.In this paper we make a significant enlargement of the class of potentials needed for a generalization of the above idea. We also make this new class amenable to construction via inverse scattering methods. We show that all one needs is a one parameter class of potentials, U(s;x), which are analytic in the strip, 0Res1, Ims>T ₀, and in addition have an asymptotic expansion in powers of [s(s–1)]^–1, i.e. U(s;x)=V ₀(x)+gV ₁(x)+g ² V ₂(x)++O(g ^N ), with g=[s(s–1)]^–1. The potentials V _n (x) are real and summable. Under suitable conditions on the V _n s and the O(g ^N ) term we show that the condition, ₀ |f ₀(x)|² V ₁(x)dx0, where f ₀ is the zero energy and g=0 Jost function for U, is sufficient to guarantee that the zeros g _n are real and, hence, s _n =1/2+i _n , for _n T ₀.Starting with a judiciously chosen Jost function, M(s,k), which is constructed such that M(s,0) is Riemann's (s) function, we have used inverse scattering methods to actually construct a U(s;x) with the above properties. By necessity, we had to generalize inverse methods to deal with complex potentials and a nonunitary S-matrix. This we have done at least for the special cases under consideration.For our specific example, ₀ |f ₀(x)|² V ₁(x)dx=0 and, hence, we get no restriction on Img _n or Res _n . The reasons for the vanishing of the above integral are given, and they give us hints on what one needs to proceed further. The problem of dealing with small but nonzero energies is also discussed.     

Anisotropy of the local field in anisotropic films of conjugated polymers

Using the data on dispersion of refractive indices in the visible region, the experimental values of the components L _j of the Lorentz tensor have been obtained for uniaxial (crystalline, oriented on a substrate, and stretched) films of conjugated polymers from nanometer to micrometer thicknesses. The dependence of the components L _j on the axial and planar orientations of macromolecules with respect to the optical axis of the film, technology of fabricating the films, and chemical structure of macromolecules has been elucidated. The correlation between the components L _j and the measured parameter η has been determined and the analytical dependence L _j (η) for films with both types of macromolecular orientation has been established. In the visible region (0 ≤ η ≤ 0.5), the presence of one point (η = 1/3) of isotropization of the tensor L and three points (η = 0, 1/3, and 0.5) of isotropization of the local field tensor f has been revealed. The point η = 1/3 of isotropization of the tensors L and f has been determined. A comparison has been made between the values of L _j (η) with predictions of known models of the local field for conjugated polymers.     

Breit–Wigner Approximation and the Distribution¶of Resonances

For operators with a discrete spectrum, {λ _j ²}, the counting function of λ _j 's, N (λ), trivially satisfies N ( λ+δ ) −N ( λ−δ ) =∑ _j δ_{λ j} ((λ−δ,λ+δ]). In scattering situations the natural analogue of the discrete spectrum is given by resonances, λ _j ∈ℂ₊, and of N (λ), by the scattering phase, s(λ). The relation between the two is now non-trivial and we prove that

Internal cooling efficiency of a junction diode

The three thermal rate equations were built newly up at both ends and at the junction of a p–n diode, in order to derive analytically the temperature difference ΔT (between a junction and both ends) and the internal cooling efficiency η defined newly for a homojunction diode. The maxima ΔT and η of a diode were derived analytically as a function of V _j within the short-length approximation and calculated numerically as a function of V _j or V _bi, where V _j is a voltage across the junction and V _bi is a built-in voltage at the junction. As a result, ΔT increases abruptly with an increase of V _j below V _j=0.050 V or of V _bi below V _bi=0.10 V, while above their values, it increases slowly with an increase of V _j or V _bi to saturate a certain value. For example, ΔT was estimated as 14.6 K for Hg_0.8Cd_0.2Te diode with V _bi=0.36 V. η has a local maximum of 63% at V _j≈0.01 V or at V _bi≈0.03 V, while above their respective values, it decreases abruptly with an increase of V _j or V _bi and falls to 4.4% at V _bi=0.80 V which is equivalent to that of a diode emitting a laser for fiber optical communication. However, the greater enhancements in ΔT and η of a diode are required to apply the internal cooling system to a laser-emitting diode which needs the exact control of temperature. These results should be useful for the application of the internal cooling system to the double heterojunction diode used in the optical communication.     

Phases of the number-theoretic spin chain

We present numerical and analytical evidence for a first-order phase transition of the ferromagnetic spin chain with partition functionZ()=(–1)/() at the inverse temperature _cr=2.     

Kraus Operator-Sum Representation and Time Evolution of Distribution Functions in Phase-Sensitive Reservoirs

Using the thermal entangled state representation 〈η|, we examine the master equation (ME) describing phase-sensitive reservoirs. We present the analytical expression of solution to the ME, i.e., the Kraus operator-sum representation of density operator ρ is given, and its normalization is also proved by using the IWOP technique. Further, by converting the characteristic function χ(λ) into an overlap between two “pure states” in enlarged Fock space, i.e., χ(λ)=〈η _=−λ |ρ|η ₌₀〉, we consider time evolution of distribution functions, such as Wigner, Q- and P-function. As applications, the photon-count distribution and the evolution of Wigner function of photon-added coherent state are examined in phase-sensitive reservoirs. It is shown that the Wigner function has a negative value when kt\leqslant\frac 12ln( 1+m_￥) \kappa t\leqslant\frac {1}{2}\ln ( 1+\mu_{\infty}) is satisfied, where μ _∞ depends on the squeezing parameter |M|² of environment, and increases as the increase of |M|.     

Application of a ‘Jacobi Identity’ for Vertex Operator Algebras to Zeta Values and Differential Operators

We explain how to use a certain new Jacobi identity for vertex operator algebras, announced in a previous paper (math.QA/9909178), to interpret and generalize S. Bloch's recent work relating values of the Riemann zeta function at negative integers to a certain Lie algebra of operators.     

Inverse problem for the Grad-Shafranov equation with affine right-hand side

For a given domain ω ⋐ ℝ² with boundary γ = ∂ω, we study the cardinality of the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) of pairs of numbers (a, b) for which there is a function u = u _(a,b): ω → ℝ such that ∇² u(x) = au(x) + b ⩾ 0 for x ∈ ω, u| _γ = 0, and ||∇u(s)| − Φ(s) ⩽ η for s ∈ γ. Here η ⩾ 0 stands for a very small number, Φ(s) = |∇(s)| / ∫ _γ |∇v| d γ, and v is the solution of the problem ∇² v = a ₀ v + 1 ⩾ 0 on ω with v| _γ = 0, where a ₀ is a given number. The fundamental difference between the case η = 0 and the physically meaningful case η > 0 is proved. Namely, for η = 0, the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) contains only one element (a, b) for a broad class of domains ω, and a = a ₀. On the contrary, for an arbitrarily small η > 0, there is a sequence of pairs (a _j , b _j ) ∈ $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) and the corresponding functions u _j such that ‖f _u ^j+1‖ − ‖f _u ^j ‖ > 1, where ‖f _u ^j = max _x∈ω |f _u ^j (x)| and f _u ^j (x) = a _j u _j (x) + b _j . Here the mappings f _u ^j : ω → ℝ necessarily tend as j → ∞ to the δ-function concentrated on γ.     

Local-field anisotropy in conjugated polymers

The experimental values of the Lorentz tensor components L _j have been obtained for uniaxial optically positive and negative films of conjugated polymers with thicknesses of tens of nanometers. The correlation between the components L _j and the measurable parameter η has been revealed and the analytical dependence L _j (η) has been derived in agreement with the experiment.     

Scale-free random graphs and Potts model

We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertexi has a prescribed weight P_i ∝ i^-μ (0 < μ< 1) and an edge can connect verticesi andj with rateP _i P _j . Corresponding equilibrium ensemble is identified and the problem is solved by theq → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ > 3 and 2 < λ < 3, whereλ = 1 +μ ^-1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finiteN shows double peaks.     

Central jet production as a probe of the perturbative formalism for exclusive diffraction

We propose a new variable, R_j, in order to identify exclusive double-diffractive high E_T di-jet production. The variable R_j is calculated using the transverse energy E_T and pseudorapidity of the jet with the largest E_T. For a purely exclusive event the value of R_j→1, if we were to neglect hadronisaton and the detector resolution effects. To illustrate the expected R_j distribution we also compute exclusive three-jet production; and, moreover, we include jet smearing effects. By studying the predictions as a function of the size of the rapidity interval, δη, which allows for additional gluon radiation, one can probe the QCD radiation effects which are responsible for the Sudakov suppression of the exclusive amplitude. In this way we may check and improve the formalism used to predict the cross sections of exclusive double-diffractive Higgs boson (and/or other new physics) production.     

Thomae Formula for General Cyclic Covers of $${{\mathbb {CP}}^1}$$

Let X be a general cyclic cover of \mathbbCP¹{\mathbb{CP}^{1}} ramified at m points, λ₁... λ _m . we define a class of non-positive divisors on X of degree g −1 supported in the pre images of the branch points on X, such that the Riemann theta function does not vanish on their image in J(X). We generalize the results of Bershadsky and Radul (Commun Math Phys 116:689–700, 1988), Nakayashiki (Publ Res Inst Math Sci 33(6):987–1015, 1997) and Enolskii and Grava (Lett Math Phys 76(2–3):187–214, 2006) and prove that up to a certain determinant of the non-standard periods of X, the value of the Riemann theta function at these divisors raised to a high enough power is a polynomial in the branch point of the curve X. Our approach is based on a refinement of Accola’s results for 3 cyclic sheeted cover (Accola, in Trans Am Math Soc 283:423–449, 1984) and a generalization of Nakayashiki’s approach explained in Nakayashiki (Publ Res Inst Math Sci 33(6):987–1015, 1997) for general cyclic covers.     

Jeans-Buneman instability in a dusty plasma

A self consistent formulation of the Jeans instability of a dusty plasma with proper inclusion of charge dynamics is described. It is shown that charge fluctuations significantly affect the Jeans as well as the Buneman mode. For plasma particles (electrons and ions) in local thermal equilibrium, the Jeans lengthλ _J is given byλ _J≈λ _g F(R, ε, β/η), whereλ _g is the Debye length of the charged grains,R is the square of the ratio of the Jeans to the plasma frequency of the grains,ε is the square of the ratio of the Debye length of the grains and the plasma particles andβ/η is the ratio of the attachment to the decay frequency of the electronic charges to the grain surface. The functional form ofF is given in the text. Numerical investigation of the Jeans-Buneman mode for a two and three component plasma shows that the Jeans mode dominates atkλ _D≪1 (wherek is the wave number andλ _D is the Debye length of plasma particles), whereas atkλ _D≫1 only the Buneman mode operates. Charge fluctuations reduce the area of overlap of the two modes. Furthermore, in the absence of gravity, there exists a new, charge fluctuation induced unstable mode in a streaming dusty plasma. Astrophysical applications of the results are discussed.     

Higher Selberg Zeta Functions

In the paper [KW2] we introduced a new type of Selberg zeta function for establishing a certain identity among the non-trivial zeroes of the Selberg zeta function and of the Riemann zeta function. We shall call this zeta function a higher Selberg zeta function. The purpose of this paper is to study the analytic properties of the higher Selberg zeta function z(s), especially to obtain the functional equation. We also describe the gamma factor of z(s) in terms of the triple sine function explicitly and, further, determine the complete higher Selberg zeta function with having a discussion of a certain generalized zeta regularization.Work in part supported by Grant-in Aid for Scientific Research (B) No.11440010, and by Grant-in Aid for Exploratory Research No.13874004, Japan Society for the Promotion of Science     

Particle correlation in superionic conductors

A molecular-dynamics method is applied to study the particle correlation in superionic conductor α-AgI. It is found that there is a close connection between the particle correlation λ_i,j (i,j=I or Ag) and the radial distribution function. The values of λ_II and λ_IAg at the nearest neighbor site are below 0.3. The values of λ_AgAg at the nearest neighbor site and λ_II at the next nearest neighbor site are about 1/2 ≈ 1/3 of it of λ_II (and λ_IAg) at the nearest neighbor site.