Numbers of the form kf(k)kf(k)

For a function $f\colon \mathbb{N}\to\mathbb{N}$, define N^{\times}_{f}(x)=\#\{n\leq x: n=kf(k) \mbox{ for some k} \}. Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and $\varphi(n)=\#\{1\leq k\leq n: (k,n)=1 \}$ be Euler's totient function. We prove that \begin{gather*} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 1) \quad N^{\times}_{\tau}(x) \asymp \frac{x}{(\log x)^{1/2}}; \\ 2) \quad N^{\times}_{\omega}(x) = (1+o(1))\frac{x}{\log\log x}; \\ \!\!\!\!\!\!\!\!\! 3) \quad N^{\times}_{\varphi}(x) = (c_0+o(1))x^{1/2}, \end{gather*} where $c_0=1.365...$\,.Comment: The error term in Theorem 1.2 is improved in this version of the pape

Estimation of KL divergence: optimal minimax rate

The problem of estimating the Kullback-Leibler divergence D(P||Q) between two unknown distributions P and Q is studied, under the assumption that the alphabet size k of the distributions can scale to infinity. The estimation is based on m independent samples drawn from P and n independent samples drawn from Q. It is first shown that there exists no consistent estimator that guarantees asymptotically small worst-case quadratic risk over the set of all pairs of distributions. A restricted set that contains pairs of distributions, with density ratio bounded by a function f(k), is further considered. An augmented plug-in estimator is proposed, and is shown to be consistent if and only if m has an order greater than k∨log^2(f(k)), and n has an order greater than kf(k). Moreover, the minimax quadratic risk is characterized to be within a constant factor of (k/(m log k)+kf(k)/(n log k))^2+log^2 f(k)/m+f(k)/n, if m and n exceed constant factors of k/log(k) and kf(k)/log k, respectively. The lower bound on the minimax quadratic risk is characterized by employing a generalized Le Cam's method. A minimax optimal estimator is then constructed by employing both the polynomial approximation and plug-in approaches

Multipolar Fermi-surface deformation in a Rydberg-dressed Fermi gas with long-range anisotropic interactions

We study theoretically the deformation of the Fermi surface (FS) of a three-dimensional gas of Rydberg-dressed 6Li atoms. The laser dressing to high-lying Rydberg D states results in angle-dependent soft-core-shaped interactions whose anisotropy is described by multiple spherical harmonics. We show that this can drastically modify the shape of the FS and that its deformation depends on the interplay between the Fermi momentum kF and the reciprocal momentum ¯k corresponding to the characteristic soft-core radius of the dressing-induced potential. When kF<¯k, the dressed interaction stretches a spherical FS into an ellipsoid. When kF≳¯k, complex deformations are encountered which exhibit multipolar characteristics. We analyze the formation of Cooper pairs around the deformed FS and show that they occupy large orbital angular momentum states (p, f, and h wave) coherently. Our study demonstrates that Rydberg dressing to high angular momentum states may pave a route toward the investigation of unconventional Fermi gases and multiwave superconductivity

Nonuniversal prefactors in correlation functions of 1D quantum liquids

We develop a general approach to calculating "nonuniversal" prefactors in static and dynamic correlation functions of 1D quantum liquids at zero temperature, by relating them to the finite size scaling of certain matrix elements (form factors). This represents a new, powerful tool for extracting data valid in the thermodynamic limit from finite-size effects. As the main application, we consider weakly interacting spinless fermions with an arbitrary pair interaction potential, for which we perturbatively calculate certain prefactors in static and dynamic correlation functions. We also non-perturbatively evaluate prefactors of the long-distance behavior of correlation functions for the exactly solvable Lieb-Liniger model of 1D bosons

Landau-Fermi liquid analysis of the 2D t-t' Hubbard model

We calculate the Landau interaction function f(k,k') for the two-dimensional t-t' Hubbard model on the square lattice using second and higher order perturbation theory. Within the Landau-Fermi liquid framework we discuss the behavior of spin and charge susceptibilities as function of the onsite interaction and band filling. In particular we analyze the role of elastic umklapp processes as driving force for the anisotropic reduction of the compressibility on parts of the Fermi surface.Comment: 10 pages, 16 figure