Related papers: Shifted convolution and the Titchmarsh divisor pro…
We study the shifted convolution problem for the divisor function in function fields in the large degree limit, that is, the average value of $d(f) d(f+h)$ where $f$ runs over monic polynomials in $\mathbb{F}_q[T]$ of a given degree, and…
The auto-correlations of arithmetic functions, such as the von Mangoldt function, the M\"obius function and the divisor function, are the subject of classical problems in analytic number theory. The function field analogues of these…
We prove a function field version of Chowla's conjecture on the autocorrelation of the Mobius function in the limit of a large finite field.
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…
We discuss a fast approximate solution to the associated classical -- classical orthogonal polynomial connection problem. We first show that associated classical orthogonal polynomials are solutions to a fourth-order quadratic eigenvalue…
In the present article, our aim is to examine some useful problems including the convolution problem, sufficiency criteria, coefficient estimates and Fekete-Szego type inequalities for a new subfamily of analytic and multivalent functions…
We investigate a function field analogue of a recent conjecture on autocorrelations of sums of two squares by Freiberg, Kurlberg and Rosenzweig, which generalizes an older conjecture by Connors and Keating. In particular, we provide…
The Titchmarsh convolution theorem is a celebrated result about the support of the convolution of two functions. We present a simple proof based on the canonical factorization theorem for bounded holomorphic functions on the unit disk.
Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field $\mathbb{F}_{q}$. For arithmetic functions $\psi_{1}, \psi_{2}: \mathbb{F}_{q}[t]\rightarrow\mathbb{C}$, we establish that if a Bombieri-Vinogradov type equidistribution…
We use the AdS/CFT correspondence to calculate CFT correlation functions of vector and spinor fields. The connection between the AdS and boundary fields is properly treated via a Dirichlet boundary value problem.
While several instances of shifted convolution problems for GL(3) x GL(2) have been solved, the case where one factor is the classical divisor function and one factor is a GL(3) Fourier coefficient has remained open. We solve this case in…
Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental…
We consider mapping properties of the iterated Stieltjes transform, establishing its new relations with the iterated Hilbert transform (a singular integral) on the half-axis and proving the corresponding convolution and Titchmarsh's type…
For elliptic in the half-space and parabolic degenerating on the boundary equation of Keldysh type we construct by similarity method the self-similar solution, which is the approximation to the identity in the class of integrable functions.…
We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field…
Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|<n\leq x} f(n) \tau(n-h)$, where $\tau$ denotes the divisor function and…
We consider conformal field theories around points of large twist degeneracy. Examples of this are theories with weakly broken higher spin symmetry and perturbations around generalised free fields. At the degenerate point we introduce twist…
In this paper, we prove both of the Titchmarsh theorems associated with Two-Sided Quaternionic Fourier Transform and we conclude about Short-Time Two-Sided Quaternionic Fourier Transform.
The frequency of occurrence of "locally repeated" values of arithmetic functions is a common theme in analytic number theory, for instance in the Erd\H{o}s-Mirsky problem on coincidences of the divisor function at consecutive integers, the…
The error function occurs widely in multiple areas of mathematics, mathematical physics and natural sciences. There has been no work in this area for the past four decades. In this article, we estimate the coefficient bounds with…