Searcharxiv⌕ Search

arXiv subjects

Farzad Aryan

Publications and source records attributed to Farzad Aryan.

7 recordsLinked to original sources
arXiv2025

Pair Correlation Factor and the Sample Complexity of Gaussian Mixtures

We study the problem of learning Gaussian Mixture Models (GMMs) and ask: which structural properties govern their sample complexity? Prior work has largely tied this complexity to the minimum pairwise separation between components, but we demonstrate this view is incomplete. We introduce the \emph{Pair Correlation Factor} (PCF), a geometric quantity capturing the clustering of component means. Unlike the minimum gap, the PCF more accurately dictates the difficulty of parameter recovery. In the uniform spherical case, we give an algorithm with improved sample complexity bounds, showing when more than the usual $ε^{-2}$ samples are necessary.

cs.LG↗
arXiv2020

Higher moments of distribution of zeta zeros

We develop a method for mean-value estimation of long Dirichlet polynomials. For an application, we use our method to study properties of the logarithmic derivative of the Riemann zeta function.

math.NT↗
arXiv2020

A new approach to gaps between zeta zeros

We study the value-distribution of Dirichlet polynomials on the critical line $\Re(s)=\tfrac{1}{2}$. As a consequence, we prove a corollary on small consecutive gaps between zeros of the Riemann zeta function. We also examine the distribution of zeros under the so-called alternative hypothesis and present a new approach to the problem of gaps between the zeros.

math.NT↗
arXiv2019

On an extension of the Landau-Gonek formula

We prove an extension of the Landau-Gonek formula. As an application we recover unconditionally some of the consequences of a pair correlation estimate that previously was known under the Riemann hypothesis. As one corollary we prove that at least two-thirds of the zeros of the zeta function are simple under a zero density hypothesis, which is weaker than the Riemann hypothesis. The results in this paper can be viewed as pair correlation estimates independent of the Riemann hypothesis.

math.NT↗
arXiv2015

Distribution of squares modulo a composite number

In this paper we study the distribution of squares modulo a square-free number $q$. We also look at inverse questions for the large sieve in the distribution aspect and we make improvements on existing results on the distribution of $s$-tuples of reduced residues.

math.NT↗