Related papers: Two identities involving Cohen-Ramanujan expansion…

Cohen-Ramanujan sum, denoted by $c_r^s(n)$, is an exponential sum similar to the Ramanujan sum $c_r(n):=\sum\limits_{\substack{h=1\\{(h,r)=1}}}^{r}e^{\frac{2\pi i n h}{r}}$. An arithmetical function $f$ is said to admit a Cohen-Ramanujan…

Some necessary and sufficient conditions for the existence of Cohen-Ramanujan expansions for arithmetical functions were provided by these authors in [\textit{arXive preprint arXive:2205.08466}, 2022]. Given two arithmetical functions $f$…

For an arithmetical function $f$, its Ramanujan expansion is a series expansion in the form $f(n)=\sum\limits_{k=1}^{\infty}a(k) c_k(n)$ where $a(k)$ are complex numbers and $c_k(n):= \sum\limits_{\substack{m=1\\(m, k)=1}}^{k}e^{\frac{2\pi…

Srinivasa Ramanujan provided series expansions of certain arithmetical functions in terms of the exponential sums defined by $c_r(n) = \sum\limits_{\substack{{m=1}\\ (m,r)=1}}^{r} e^{\frac{2 \pi imn}{r}}$ in [Trans. Cambridge Phillos. Soc,…

For two arithmetical functions $f$ and $g$, we study the convolution sum of the form $\sum_{n \le N} f(n) g(n+h)$ in the context of its asymptotic formula with explicit error terms. Here we introduce the concept of finite Ramanujan…

Srinivasa Ramanujan provided Fourier series expansions of certain arithmetical functions in terms of the exponential sum defined by $c_q(n)=\sum\limits_{\substack{{m=1}\\(m,q)=1}}^{q}e^{\frac{2 \pi imn}{q}}$. Later, H. Delange derived the…

We continue our study of convolution sums of two arithmetical functions $f$ and $g$, of the form $\sum_{n \le N} f(n) g(n+h)$, in the context of heuristic asymptotic formul\ae. Here, the integer $h\ge 0$ is called, as usual, the {\it shift}…

All arithmetical functions $F$ satisfying Ramanujan Conjecture, i.e., $F(n)\ll_{\varepsilon}n^{\varepsilon}$, and with $Q-$smooth divisors, i.e., with Eratosthenes transform $F':=F\ast \mu$ supported in $Q-$smooth numbers, have a kind of…

Given two arithmetical functions $f,g$ we derive, under suitable conditions, asymptotic formulas with error term, for the convolution sums $\sum_{n \le N} f(n) g(n+h)$, building on an earlier work of Gadiyar, Murty and Padma. A key role in…

For two arithmetical functions $f$ and $g$ with absolutely convergent Ramanujan expansions, Murty and Saha have recently derived asymptotic formulas with error term for the convolution sum $\sum_{n \le N} f(n) g(n+h)$ under some suitable…

We define an extension of the Ramanujan trigonometric function to arbitrary dimensions, and give the Dirichlet series generating function. The extension was first given by Eckford Cohen long ago. This links directly to visible point vector…

Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field $\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in $\mathbb{A}$. In this paper, we show that a large class of arithmetic functions in…

Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion…

Using Abel summation the paper proves a weak form of the Wiener-Khinchin formula for arithmetic functions with point-wise convergent Ramanujan-Fourier expansions. The main result is that the convolution of most arithmetic functions…

In this self-contained short note, we prove that {\it every arithmetic function} $F$ {\it has infinitely many Ramanujan coefficients} $G$ {\it giving an absolutely convergent Ramanujan expansion for $F$}. This is "coefficients'…

Let $c_q(n)$ denote the Ramanujan sum modulo $q$, and let $x$ and $y$ be large reals, with $x = o(y)$. We obtain asymptotic formulas for the sums $$\sum_{n \le y}(\sum_{q \le x} c_q(n))^k \qquad (k = 1, 2).$$

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…

Ramanujan sums are exponential sums with exponent defined over the irreducible fractions. Until now, they have been used to provide converging expansions to some arithmetical functions appearing in the context of number theory. In this…

In this paper, we give some extensions for Ramanujan's circular summation formula with the mixed products of two Jacobi's theta functions. As some applications, we also obtain many interesting identities of Jacobi's theta functions.

In the work we shall present formulas to sum Lambert series using Euler's q-exponential functions, and several Lambert series associated with well-known arithmetic functions are given as examples. These functions are: the M\"{o}bius…