The fourth moment of Dirichlet L-functions

In this paper, we study the fourth moment of Dirichlet $L$-functions averaged over primitive characters to modulus $q$ and over $t\in [0,T]$, for general $q$ and $T$. When $T\gg q^\varepsilon$ is not too small with respect to $q$, we obtain an asymptotic formula with a power savings in the error term. In addition, when $q$ is a prime, the weak condition $T\gg q^\varepsilon$ is unnecessary, as well as a more sharp error has been obtained. These improvements also benefit from an extension of Young's classical work [The fourth moment of Dirichlet L-functions, Ann. of Math. (2) 173 (2011), no. 1, 1-50] for prime $q$.Comment: 63 pages, any comments are welcome. We fix some minor typos in English in this versio

ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is three-fold: We firstly develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implicates that shearlet theory provides a unified treatment of both the continuum and digital realm. Secondly, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying Matlab toolbox called ShearLab is provided. And, thirdly, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.Comment: submitted to SIAM J. Multiscale Model. Simu

On gaps between zeros of the Riemann zeta function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they differ by at most 0.5154 times the average spacing.Comment: submitted for publication in January 201