On the special values of certain L-series related to half-integral weight modular forms

Let h be a cuspidal Hecke eigenform of half-integral weight, and En/2+1/2 be Cohen’s Eisenstein series of weight n/2+1/2. For a Dirichlet character χ we define a certain linear combination R(χ)(s, h,En/+1/2) of the Rankin-Selberg convolution products of h and En/2+1/2 twisted by Dirichlet characters related with χ. We then prove a certain algebraicity result for R(χ)(l, h,En/2+1/2) with l integers

Interacting quantum rotors in oxygen-doped germanium

We investigate the interaction effect between oxygen impurities in crystalline germanium on the basis of a quantum rotor model. The dipolar interaction of nearby oxygen impurities engenders non-trivial low-lying excitations, giving rise to anomalous behaviors for oxygen-doped germanium (Ge:O) below a few degrees Kelvin. In particular, it is theoretically predicted that Ge:O samples with oxygen-concentration of 10$^{17-18}$cm$^{-3}$ show (i) power-law specific heats below 0.1 K, and (ii) a peculiar hump in dielectric susceptibilities around 1 K. We present an interpretation for the power-law specific heats, which is based on the picture of local double-well potentials randomly distributed in Ge:O samples.Comment: 13 pages, 11 figures; to be published in Phys. Rev.

CeCu_2Ge_2: Challenging our Understanding of Quantum Criticality

Here, we unveil evidence for a quantum phase-transition in CeCu_2Ge_2 which displays both an incommensurate spin-density wave (SDW) ground-state, and a strong renormalization of the quasiparticle effective masses (mu) due to the Kondo-effect. For all angles theta between an external magnetic field (H) and the crystallographic c-axis, the application of H leads to the suppression of the SDW-state through a 2^nd-order phase-transition at a theta-dependent critical-field H_p(theta) leading to the observation of small Fermi surfaces (FSs) in the paramagnetic (PM) state. For H || c-axis, these FSs are characterized by light mu's pointing also to the suppression of the Kondo-effect at H_p with surprisingly, no experimental evidence for quantum-criticality (QC). But as $H$ is rotated towards the a-axis, these mu's increase considerably becoming undetectable for \theta > 56^0 between H and the c-axis. Around H_p^a~ 30 T the resistivity becomes proportional T which, coupled to the divergence of mu, indicates the existence of a field-induced QC-point at H_p^a(T=0 K). This observation, suggesting FS hot-spots associated with the SDW nesting-vector, is at odds with current QC scenarios for which the continuous suppression of all relevant energy scales at H_p(theta,T) should lead to a line of quantum-critical points in the H-theta plane. Finally, we show that the complexity of its magnetic phase-diagram(s) makes CeCu_2Ge_2 an ideal system to explore field-induced quantum tricritical and QC end-points.Comment: 10 pages, 5 figures, Phys. Rev. B (in press

Restricted sum formula of alternating Euler sums

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Functional central limit theorems for vicious walkers

We consider the diffusion scaling limit of the vicious walker model that is a system of nonintersecting random walks. We prove a functional central limit theorem for the model and derive two types of nonintersecting Brownian motions, in which the nonintersecting condition is imposed in a finite time interval $(0,T]$ for the first type and in an infinite time interval $(0,\infty)$ for the second type, respectively. The limit process of the first type is a temporally inhomogeneous diffusion, and that of the second type is a temporally homogeneous diffusion that is identified with a Dyson's model of Brownian motions studied in the random matrix theory. We show that these two types of processes are related to each other by a multi-dimensional generalization of Imhof's relation, whose original form relates the Brownian meander and the three-dimensional Bessel process. We also study the vicious walkers with wall restriction and prove a functional central limit theorem in the diffusion scaling limit.Comment: AMS-LaTeX, 20 pages, 2 figures, v6: minor corrections made for publicatio