Related papers: An elementary property of correlations

Hafner and Stopple proved a conjecture of Zagier, that the inverse Mellin transform of the symmetric square $L$-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the non-trivial zeros of the Riemann…

This paper aims to show that by making use of Ramanujan's Master Theorem and the properties of the lower incomplete gamma function, it is possible to construct a finite Mellin transform for the function $f(x)$ that has infinite series…

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 method of proof of Balog and Ruzsa and the large sieve of Linnik are used to investigate the behaviour of the $L^{1}$ norm of a wide class of exponential sums over the square-free integers and the primes. Further, a new proof of the…

The aim in this note is to provide a generalization of an interesting entry in Ramanujan's Notebooks that relate sums involving the derivatives of a function Phi(t) evaluated at 0 and 1. The generalization obtained is derived with the help…

For functions $f(z)= z+ a_2 z^2 + a_3 z^3 + \cdots$ in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional $\phi(f,n,m;\lambda):=|\lambda a_n a_m…

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g), n\geq 2$, \Delta_p u + a(x)u^{p-1} = \lambda f(u,x), (E2) where $\Delta_p$ is the $p-$laplacian, with $1<p<n$. The…

We construct a uniformly discrete, and even sparse, sequence of real numbers $\Lambda=\{\lambda_n\}$ and a function g in $L^2(R)$, such that for every q>2, every function f in $L^2(R)$ can be approximated with arbitrary small error by a…

The \emph{flat deformation theorem} states that given a semi-Riemannian analytic metric $g$ on a manifold, locally there always exists a two-form $F$, a scalar function $c$, and an arbitrarily prescribed scalar constraint depending on the…

For integer $q$, let $\chi$ be a primitive multiplicative character$\pmod q.$ For integer $a$ coprime to $q$, we obtain a new bound for the sums $$\sum_{n\le N}\Lambda(n)\chi(n+a),$$ where $\Lambda(n)$ is the von Mangoldt function. This…

We establish several results concerning the expected general phenomenon that, given a multiplicative function $f:\mathbb{N}\to\mathbb{C}$, the values of $f(n)$ and $f(n+a)$ are "generally" independent unless $f$ is of a "special" form.…

In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan $R$-function) $R(a)$, by showing some monotonicity, concavity and convexity properties…

In this paper, we study sums of shifted products $\sum\limits_{n \leq x} F(n) G(n-h)$ for any $|h| \leq x/2$ and arithmetic functions $F=f*1$ and $G=g*1$, with $f$ and $g$ small. We obtain asymptotic formula for different orders of…

We present a smooth version of Landaus explicit formula for the von Mangoldt arithmetical function. Assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number is a prime number, it is…

Consider the regular Dirichlet extension $(\mathcal{E},\mathcal{F})$ for one-dimensional Brownian motion, that $H^1(\mathbb{R})$ is a subspace of $\mathcal{F}$ and $\mathcal{E}(f,g)=\frac12\mathbf{D}(f,g)$ for $f,g\in H^1(\mathbb{R})$. Both…

In this paper, we provide novel mean value estimates for exponential sums related to the extended main conjecture of Vinogradov's mean value theorem, by developing the Hardy-Littlewood circle method together with a refined shifting…

Let $ \Lambda $ denote von Mangoldt's function, and consider the averages \begin{align*} A_N f (x) &=\frac{1}{N}\sum_{1\leq n \leq N}f(x-n)\Lambda(n) . \end{align*} We prove sharp $ \ell ^{p}$-improving for these averages, and sparse bounds…

We introduce new analogues of the Ramanujan sums, denoted by $\widetilde{c}_q(n)$, associated with unitary divisors, and obtain results concerning the expansions of arithmetic functions of several variables with respect to the sums…

We consider the asymptotic expansion of the functional series \[S_{\mu}^\pm(a;\lambda)=\sum_{n=0}^\infty \frac{(\pm 1)^n e^{-\lambda n}}{(n^2+a^2)^\mu}\] for $\lambda>0$ and $\mu\geq0$ as $|a|\to \infty$ in the sector $|\arg\,a|<\pi/2$. The…

Let $F$ be a function in the Selberg class ${\mathcal S}$ and $a$ be a real number not equal to 1/2. Consider the sum $$\lambda_{F}(n,a)=\sum_{\rho}\left[1-\left(\frac{\rho-a}{\rho+a-1}\right)^{n}\right],$$ where $\rho$ runs over the…