Related papers: Arithmetic correlations over large finite fields

We establish a connection between analytic number theory and computational learning theory by showing that the M\"obius function belongs to a class of functions that is statistically hard to learn from random samples. Let $\mu_R$ denote the…

We study the correlation functions of local operators in unitary $\textrm{T}\bar{\textrm{T}}$-deformed field theories, using their formulation in terms of Jackiw-Teitelboim gravity. The position of the operators is defined using the…

Through an explicit construction, we assign to any infinite temperature autocorrelation function $C(t)$ a set of functions $\alpha^n(t)$. The construction of $\alpha^n(t)$ from $C(t)$ requires the first $2n$ temporal derivatives of $C(t)$…

A motivation comes from {\em M. Ismail and et al.: A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77--84} to study a generalization of close-to-convex functions by means of a $q$-analog of a difference…

We give a differentially closed description of the uniformizing representation to the analytical apparatus on Riemann surfaces and orbifolds of finite analytic type. Apart from well-known automorphic functions and Abelian differentials it…

A correlation function of the classical orthogonal polynomials is defined and determined. The correlation function obeys a second order difference equation in two variables. The correlation function for the Gegenbauer, Chebyshev and…

We present examples of holomorphic functions that vanish to in- finite order at points at the boundary of their domain of definition. They give rise to examples of Dirichlet minimizing Q-valued functions indicating that "higher"-regularity…

By using the $q$-analogue of van der Corput's method we study the divisor function in an arithmetic progression to modulus $q$. We show that the expected asymptotic formula holds for a larger range of $q$ than was previously known, provided…

The simplest examples of chaotic maps are linear, area-preserving maps on the circle, torus, or product of tori; respectively known as the Bernoulli map, the cat map, and the recently introduced "spatiotemporal" cat map. We study…

Correlation functions in the restricted primitive model are calculated within a field-theoretic approach in the one-loop self-consistent Hartree approximation. The correlation functions exhibit damped oscillatory behavior as found before in…

Let F be a class of functions with the uniqueness property: if a function f in F vanishes on a set of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a…

We study logarithmically averaged binary correlations of bounded multiplicative functions $g_1$ and $g_2$. A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever $g_1$ or…

We study higher uniformity properties of the von Mangoldt function $\Lambda$, the M\"obius function $\mu$, and the divisor functions $d_k$ on short intervals $(x,x+H]$ for almost all $x \in [X, 2X]$. Let $\Lambda^\sharp$ and $d_k^\sharp$ be…

We study higher uniformity properties of the M\"obius function $\mu$, the von Mangoldt function $\Lambda$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed…

We study $(m)$-type connected correlation functions of OPE blocks with respect to one spatial region in two dimensional conformal field theory. We find logarithmic divergence for these correlation functions. We justify the logarithmic…

Andrews and Bressoud, Alladi and Gordon, and others, have proven, in a number of papers, that the coefficients in various arithmetic progressions in the series expansions of certain infinite $q$-products vanish. In the present paper it is…

We prove that all algebraic relations over $\overline{\mathbb Q}$ between values of Siegel's $E$-functions at some non-zero algebraic point have a functional source, in that they can be obtained as degeneration of $\delta$-algebraic…

Let $(M,g)$ be a $n-$dimensional, compact Riemannian manifold. We define the frequency scale $\lambda$ of a function $f \in C^{0}(M)$ as the largest number such that $\left\langle f, \phi_k \right\rangle =0$ for all Laplacian eigenfunctions…

In this article we study the asymptotic behaviour of the correlation functions over polynomial ring $\mathbb{F}_q[x]$. Let $\mathcal{M}_{n, q}$ and $\mathcal{P}_{n, q}$ be the set of all monic polynomials and monic irreducible polynomials…

The lecture, given at the ICM 2014 in Seoul, explores several problems of analytic number theory in the context of function fields over a finite field, where they can be approached by methods different than those of traditional analytic…