ﻻ يوجد ملخص باللغة العربية
We consider very general random integers and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle packings, trace of Frobenius elements for elliptic curves, the Ramanujan tau-function, Mersenne numbers, and many others.
We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.
Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of gi
We prove a new upper bound on the second moment of Maass form symmetric square L-functions defined over Gaussian integers. Combining this estimate with the recent result of Balog-Biro-Cherubini-Laaksonen, we improve the error term in the prime geodesic theorem for the Picard manifold.
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic num
Rudnick and Wigman (Ann. Henri Poincar{e}, 2008; arXiv:math-ph/0702081) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the $d$-dimensional torus is $O(E/mathcal{N})$, as $Etoinfty$, where $E$ is the energy