Riemann Ξ-function Research Articles

Linear law for the logarithms of the Riemann periods at simple critical zeta zeros

Each simple zero 1 2 + i γ n \frac {1}{2}+i\gamma _n of the Riemann zeta function on the critical line with γ n > 0 \gamma _n > 0 is a center for the flow s ˙ = ξ ( s ) \dot {s}=\xi (s) of the Riemann xi function with an associated period T n T_n . It is shown that, as γ n → ∞ \gamma _n \rightarrow \infty , \[ log ⁡ T n ≥ π 4 γ n + O ( log ⁡ γ n ) . \log T_n\ge \frac {\pi }{4}\gamma _n+O(\log \gamma _n). \] Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture γ n + 1 − γ n ≫ γ n − θ \gamma _{n+1}-\gamma _n \gg \gamma _n^{-\theta } for some exponent θ > 0 \theta >0 , we obtain the upper bound log ⁡ T n ≪ γ n 2 + θ \log T_n \ll \gamma ^{2+\theta }_n . Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, log ⁡ T n = π 4 γ n + O ( log ⁡ γ n ) \log T_n = \frac {\pi }{4}\gamma _n+O(\log \gamma _n) . Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert–Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis.

New Raman method for aqueous solutions: ξ-function dispersion evidence for strong F−-water H–bonds in aqueous CsF and KF solutions

The Raman xi-function dispersion method recently elucidated for the strong H-bond breaker, ClO4-, in water [G. E. Walrafen, J. Chem. Phys. 122, 094510 (2005)] is extended to the strongly H-bond forming ion, F-. Measuring the xi function is analogous to measuring DeltaG from the thermodynamic activity of the water, aH2O, as the stoichiometric mol fraction of the water in the solution decreases due to addition of an electrolyte or nonelectrolyte. xi is the derivative of the OH-stretching part of the Gibbs free energy with respect to the water mol fraction; xiomega identical with-RT[ partial differential ln(Iomega/IREF) partial differentialX2](T,P). I is the Raman intensity at omega (omega=Raman shift in cm-1); IREF, that at an arbitrary reference omega; and, X2 is the water mol fraction (X1=CsF or KF mol fraction). ln(Iomega/IREF) was found to be linear in X2 for the complete range of OH-stretching omega's, with correlation coefficients as large as 0.999 96. Linearity of ln(Iomega/IREF) versus X2 is an experimental fact for all omega's throughout the spontaneous Raman OH-stretching contour; this fact cannot be negated by relative contributions of ultrafast/fast, homogeneous/inhomogeneous processes which may underlie this linearity. Linearity in ln(Iomega/IREF) versus 1T, or in ln(Iomega/IREF) versus P, was also observed for the Raman H-bond energy DeltaE and pair volume DeltaV dispersions, respectively. A low-frequency maximum (MAX) and a high-frequency minimum (MIN) were observed in the xi function dispersion curve. Deltaxi=xiMIN-xiMAX values of -7000+/-800-cal/mol H2O for CsF, and the experimentally equal Deltaxi=-6400+/-1000-cal/mol H2O for KF, were obtained. These Deltaxi's are opposite in sign but have nearly the same absolute magnitude as the Deltaxi value for NaClO4 in water; Deltaxi=+8050+/-100-cal/mol H2O. A positive Deltaxi corresponds to a water-water H-bond breaker; a negative Deltaxi to a H-bond former; specifically, a F--water H-bond former, in the instant case. NaClO4 breaks water-water H-bonds and also gives rise to weak, long (3.0-3.3 A), severely bent (approximately 140 degrees), high-energy, ClO4--water interactions. Fluoride ion scavenges the extremely weak or non-hydrogen-bonded OH groups, thus forming strong, short, linear, low-energy, H-bonds between F- and water. The strength of the F--water H-bond is evident from the fact that the OH-stretching xi-function minimum is centered approximately 200-300 cm-1 below that of ice. The diagnostic feature of the Raman spectrum from F- in water is an intense, long, low-frequency OH-stretching tail extending 800 cm-1 or more below the 3300-cm-1 peak. A similar intense, long, low-frequency Raman tail is produced by the OH- ion, which is known to H-bond very strongly when protons from water are donated to its oxygen atom.

Toward Verification of the Riemann Hypothesis: Application of the Li Criterion

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λk}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence ηj are valid. The constants ηj enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λn and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.

The holomorphic flow of Riemann's function ξ(z)

The holomorphic flow of Riemann's xi function is considered. Phase portraits are plotted and the following results, suggested by the portraits, proved: all separatrices tend to the positive and/or negative real axes. There are an infinite number of crossing separatrices. In the region between each pair of crossing separatrices—a band—there is at most one zero on the critical line. All zeros on the critical line are centres or have all elliptic sectors. The flows for ξ(z) and cosh(z) are linked with a differential equation. Simple zeros on the critical line and Gram points never coincide. The Riemann hypothesis is equivalent to all zeros being centres or multiple together with the non-existence of separatrices which enter and leave a band in the same half plane.

Zero spacing distributions for differenced L-functions

The paper studies the local zero spacings of deformations of the Riemann ξ-function under certain averaging and differencing operations. For real h we consider the entire functions Ah(s) := 1 (ξ(s + h) + ξ(s − h)) and Bh(s) = 1 2i (ξ(s + h) − ξ(s − h)) . For |h| ≥ 1 2 the zeros of Ah(s) and Bh(s) all lie on the critical line ℜ(s) = 1 and are simple zeros. The number of zeros of these functions to height T has asymptotically the same density as the Riemann zeta zeros. For fixed |h| ≥ 1 the distribution of normalized zero spacings of these functions up to height T converge as T → ∞ to a limiting distribution, which consists of equal spacings of size 1. That is, these zeros are asymptotically regularly spaced. Assuming the Riemann hypothesis, the same properties hold for all nonzero h. In particular, these averaging and differencing operations destroy the (conjectured) GUE distribution of the zeros of the ξ-function, which should hold at h = 0. Analogous results hold for all completed Dirichlet L-functions ξχ(s) having χ a primitive character.

Correction to: ``On a positivity property of the Riemann ξ-function'' (Acta Arith. 89 (1999), 217–234)

Correction to: ``On a positivity property of the Riemann ξ-function'' (Acta Arith. 89 (1999), 217–234)

Measure concentration for Euclidean distance in the case of dependent random variables

Let qn be a continuous density function in n-dimensional Euclidean space. We think of qn as the density function of some random sequence Xn with values in $\Bbb{R}^{n}$. For I⊂[1,n], let XI denote the collection of coordinates Xi, i∈I, and let $\overline X_{I}$ denote the collection of coordinates Xi, i∉I. We denote by $Q_{I}(x_{I}|\bar{x}_{I})$ the joint conditional density function of XI, given $\overline X_{I}$. We prove measure concentration for qn in the case when, for an appropriate class of sets I, (i) the conditional densities $Q_{I}(x_{I}|\bar{x}_{I})$, as functions of xI, uniformly satisfy a logarithmic Sobolev inequality and (ii) these conditional densities also satisfy a contractivity condition related to Dobrushin and Shlosman’s strong mixing condition.

A Generalization of Newman’s Result on the Zeros of Fourier Transforms

In this paper, the class of complex Borel measures μ, satisfying μ(−E) = μ(E) for every Borel set E ⊂ ℝ, such that the functions f μ,λ, λ > 0, defined by $$f_{\mu,\lambda}(z)=\int_{-\infty}^{\infty}{\rm exp}(-{\lambda\over 2} t^2 + izt)\ d\mu(t)$$ , have only real zeros, is completely determined. It is done by establishing a general theorem (Theorem 1.3) on the asymptotic behavior of the zero-distribution of f μλ} for λ → ∞. The theorem is applied to the Riemann ξ-function.

Relations and positivity results for the derivatives of the Riemann ξ function

We present and evaluate the integer-order derivatives of the Riemann xi function. These derivatives contain logarithmic integrals of powers multiplying a specific Jacobi theta function and as such can be alternatively viewed as certain Mellin transforms at integer argument. We describe how the derivatives at s=0, s= 1 2 , and s=1 can be evaluated exactly. We further show, based upon a novel representation, that the even order derivatives at s= 1 2 are all positive, as are all derivatives at s=1. An expression is presented for the derivatives on the critical line, which may be useful in studying the zeros of the function Ξ(t)=ξ( 1 2 + it) .

A remark on the distribution of the zeros of Riemann's zeta function and a continuous analogue of Kakeya's theorem

A certain new symmetric representation of Riemann's xi function is considered. A theorem on the zeros of trigonometric integrals analogous to Kakeya's theorem on the zeros of polynomials with monotonically non-decreasing coefficients is used. A modification of Polya's method is suggested, which allows one to obtain new assertions on the disposition of the zeros of the zeta function.

Localization length in anderson insulator with Kondo impurities.

The localization length, xi, in a two-dimensional Anderson insulator depends on the electron spin scattering rate by magnetic impurities, tau(-1)(s). For antiferromagnetic sign of the exchange, the time tau(s) is itself a function of xi, due to the Kondo correlations. We demonstrate that the unitary regime of localization is impossible when the concentration of magnetic impurities, n(M), is smaller than a critical value, n(c). For n(M)>n(c), the dependence of xi on the dimensionless conductance, g, is reentrant, crossing over to unitary, and back to orthogonal behavior upon increasing g. Sensitivity of Kondo correlations to a weak parallel magnetic field results in a giant parallel magnetoresistance.

A Formula for the Dedekind ξ-Function of an Imaginary Quadratic Field

In this paper a formula is given for the Dedekind ξ-function, which is similar to that for the Riemann ξ-function. As an application, a lower bound is obtained for the value of the Dedekind ξ-function at the central point.

Conjectures and Theorems in the Theory of Entire Functions

Motivated by the recent solution of Karlin's conjecture, properties of functions in the Laguerre–Polya class are investigated. The main result of this paper establishes new moment inequalities for a class of entire functions represented by Fourier transforms. The paper concludes with several conjectures and open problems involving the Laguerre–Polya class and the Riemann ξ-function.

Exact numerical calculation of the density of states of the fluctuating gap model

We develop a powerful numerical algorithm for calculating the density of states rho(omega) of the fluctuating gap model, which describes the low-energy physics of disordered Peierls and spin-Peierls chains. We obtain rho(omega) with unprecedented accuracy from the solution of a simple initial value problem for a single Riccati equation. Generating Gaussian disorder with large correlation length xi by means of a simple Markov process, we present a quantitative study of the behavior of rho (omega) in the pseudogap regime. In particular, we show that in the commensurate case and in the absence of forward scattering the pseudogap is overshadowed by a Dyson singularity below a certain energy scale omega^{ast}, which we explicitly calculate as a function of xi.

Correlations of galaxies at Z = 0.3

The angular correlation functions for galaxies with magnitudes 20 less than b(sub J) less than 21.5 and 21 less than b(sub J) less than 22.5 are measured from photographic images of nine distinct fields. New expressions for the variance of correlation function estimates are used, which show that the nine fields are consistent with a single angular correlation function w(theta) and with previous estimates of its amplitude and slope -- though the errors on previously published w(theta) values are probably underestimated. Assuming a spatial correlation function of the form xi(r,z) = (r/r(sub 0))(exp -gamma)(1 + z)(exp -3-epsilon), and using the measured N(z) for these magnitude ranges, we derive xi(r = 250 kpc/h, z = 0.18) = 48 +/- 10, and xi(r = 250 kpc/h, z = 0.27) = 55(+11; -9) for galaxies in these samples, largely independent of gamma and epsilon (1 sigma errors). This is approximately 2 times lower than the small-scale clustering seen in nearby galaxies at b(sub J) less than 18 (after allowing for modest growth in clustering since z = 0.27), yet very similar to the clustering of the nearby IRAS-selected galaxy population. The correlations of the reddest third of the b(sub J) approximately 22 galaxies, however, are larger and consistent with the b(sub J) less than 18 clustering, not the IRAS clustering, for a clustering growth rate of epsilon equal to or greater than -1. Thus to b(sub J) less than 22.5, most of the apparent evolution in clustering of blue-selected galaxies appears to be due to an increasing fraction of late-type or star-forming galaxies at fainter magnitudes, combined with weaker small-scale clustering for late types. These conclusions are entirely empirical and independent of galaxy evolution models.

Lehmer pairs of zeros, the de Bruijn-Newman constant ?, and the Riemann Hypothesis

We give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH0 (which is closely related to the Riemann ξ-function) to be a “Lehmer pair” of zeros ofH0. With this formulation, we establish that each such pair of zeros gives a lower bound for the de Bruijn-Newman constant Λ (where the Riemann Hypothesis is equivalent to the assertion that Λ≤0). We also numerically obtain the following new lower bound for Λ: $$ - 4.379 \cdot 10^{ - 6}< \Lambda $$

The GHS inequality and the Riemann hypothesis

LetV(t) be the even function on (−∞, ∞) which is related to the Riemann xi-function by Ξ(x/2)=4∫−∞∞ exp(ixt−V(t))dt. In a proof of certain moment inequalities which are necessary for the validity of the Riemann Hypothesis, it was previously shown thatV'(t)/t is increasing on (0, ∞). We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely thatV' is convex on [0, ∞). The possible relevance of the convexity ofV' to the Riemann Hypothesis is discussed.

Jensen polynomials with applications to the Riemann ξ-function

In this paper, generalizations of some known results for Jensen polynomials, pertaining to (i)convexity, (ii) the Turán inequalities, and (iii) the Laguerre inequalities, are established. These results are then applied in general to real entire functions, which are representable by Fourier transforms, and in particular to the Riemann ξ-function.

A numerical verification of the Clausius-Mossotti relation

The results of the approximation formula deduced by Clausius and Mossotti to calculate the effective permittivity of an arrangement of a large number of spheres with a permittivity xi /sub s/ filling a medium with a permittivity xi /sub s/ are compared with a highly accurate three-dimensional finite-element method solution. This formula gives the effective permittivity as a function of xi /sub s/ and sigma /sub m/ as well as the ratio of the two volumes. The influence of different structures with the same volume ratios is investigated. It is shown that there is only an insignificant difference between the Clausius-Mossotti approximation formula and the highly accurate finite-element solution for different lattice structures. The differences between the two solutions become more pronounced for higher values of the volume ratio and of the conductivity ratio of the two materials.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Constraints on large-scale clustering from the autocorrelation properties of the X-ray background

Expectations for the angular correlation function of intensity fluctuations of the extragalacitc 2-10 keV X-ray background are discussed in relation to the two-point spatial correlation functions, xi(r), of X-ray sources. A simple analytic formula for the amplitude of intensity correlations, Gamma(theta), holding in the low-redshift, small-separation approximation, is derived. The HEAO 1 A-2 upper limits on Gamma(theta) have been exploited to derive constraints on the local xi(r) functions of rich clusters of galaxies, of AGNs, and on the AGN-cluster cross-correlation, as well as on evolution of the correlation functions with cosmic time. X-ray data are found to be compatible with the Bahcall and Soneira estimates of xi(cluster-cluster) as well as with the results of recent studies of quasar clustering.