Related papers: On the Gauss Circle Problem
Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for…
The Gauss circle problem concerns with the evaluation of $\sum_{n \leq x}r(n)$, where $r(n)$ denotes the number of representations of $n$ as sums of two squares and $x \geq 2$. Let $\Psi_G(x,y)$ denote the sum of $y$-smooth numbers below…
We study a number of possible extensions of the Ramanujan master theorem, which is formulated here by using methods of Umbral nature. We discuss the implications of the procedure for the theory of special functions, like the derivation of…
We give an asymptotic formula for the mean value of the number of representations of an integer as sum of two squares known as the Gauss circle problem.
In this note, we solve the Gauss image problem given two Borel measures on the unit sphere, one of which is absolutely continuous with respect to the uniform measure.
A new representation for a regular solution of the radial Dirac system of a special form is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to the spectral parameter. For the…
Using Bombieri-Iwaniec method, we establish an improvement on both Gauss's Circle Problem and Dirichlet's Divisor Problem. More precisely, we derive a new estimate for the first spacing problem and combine it with Huxley's results on the…
The q-Laguerre polynomials correspond to an indetermined moment problem. For explicit discrete non-N-extremal measures corresponding to Ramanujan's ${}_1\psi_1$-summation we complement the orthogonal q-Laguerre polynomials into an explicit…
In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an "extra" parameter, and…
In this paper, we establish the irrationality of some open problems in mathematics based on using a recursive formula that generate the complete sequence of numbers. see [1] But before getting into that we begin with some Ramanujan notable…
In this short research note, we aim to establish an interesting extension of a summation due to Ramanujan.The result is derived with the help of an extension of Gauss's summation theorem available in the literature.
The general properties of two-dimensional generalized Bessel functions are discussed. Various asymptotic approximations are derived and applied to analyze the basic structure of the two-dimensional Bessel functions as well as their nodal…
We define bilateral series related to Ramanujan-like series for $1/\pi^2$. Then, we conjecture a property of them and give some applications.
This paper is devoted to the proof Gauss' divergence theorem in the framework of "ultrafunctions". They are a new kind of generalized functions, which have been introduced recently [2] and developed in [4], [5] and [6]. Their peculiarity is…
We report on a new class of exact solutions of the scalar Helmholtz equation obtained by carefully engineering the form of the angular spectrum of a Bessel beam. We consider in particular the case in which the angular spectrum of such…
A new representation for a regular solution of the perturbed Bessel equation of the form $Lu=-u"+\left( \frac{l(l+1)}{x^2}+q(x)\right)u=\omega^2u$ is obtained. The solution is represented as a Neumann series of Bessel functions uniformly…
The lattice point problems of the $p$-circle (for example, the astroid), which a generalized circle for positive real numbers $p$, have been solved for approximately $p$ more than 3, based on the series representation of the error term…
We study Ramanujan-Fourier series of certain arithmetic functions of two variables. We generalize Delange's theorem to the case of arithmetic functions of two variables and give sufficient conditions for pointwise convergence of…
We prove that there is a correspondence between Ramanujan-type formulas for 1/\pi, and formulas for Dirichlet L-values. The same method also allows us to resolve certain values of the Epstein zeta function in terms of rapidly converging…
Ramanujan derived the well known divergent-sum of integers in more than one way. We generalise the informal method to higher powers of the Riemann zeta function through a study of the Eulerian numbers in particular. Within the context of…