Related papers: Sharp Approximations for the Ramanujan Constant

Let $V\subset\R^m$ be a centrally symmetric convex body and let $V^*\subset\R^m$ be its polar. We prove limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials…

By means of a variational approach we find new series representations both for well known mathematical constants, such as $\pi$ and the Catalan constant, and for mathematical functions, such as the Riemann zeta function. The series that we…

The aim of this paper is to study the convergence and divergence of the Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on the unit circle. We provide an example of an uncountable set of measure zero on which the…

In this paper, we study some properties such as the monotonicity, logarithmically complete monotonicity, logarithmic convexity, and geometric convexity, of the combinations of gamma function and power function. The results we obtain…

We develop approximations for the Riemann zeta function that enable high-precision computation within the critical strip and other vertical strips. These approximations combine the main sum of the Riemann-Siegel formula with a simple…

We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of…

We give an elementary proof for new strict upper and lower bounds for the correction term in Ramanujan's approximation for the factorial function

A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of…

In the paper, the authors review origins, motivations, and generalizations of a series of inequalities involving several exponential functions and sums, establish three new inequalities involving finite exponential functions and sums by…

We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.

In this paper, some monotonicity and concavity results of several functions involving the psi and polygamma functions are proved, and then some known inequalities are extended and generalized.

The Ramanujan Machine project detects new expressions related to constants of interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures concerning the…

In this article we use theoretical and numerical methods to evaluate in a closed-exact form the parameters of Ramanujan type $1/\pi$ formulas.

In this paper, we prove that the double inequality \begin{equation*} 1+\alpha r'^2<\frac{\mathcal{K}_{a}(r)}{\sin(\pi a)\log(e^{R(a)/2}/r')}<1+\beta r'^2 \end{equation*} holds for all $a\in (0, 1/2]$ and $r\in (0, 1)$ if and only if…

The goal of this paper is to formulate a systematical method for constructing the fastest possible continued fraction approximations of a class of functions. The main tools are the multiple-correction method, the generalized Mortici's lemma…

We have found several summation formulas that extend Ramanujan's psi sum. First contains a parameter $\alpha=1/N$, $N$ is a positive integer, and transforms to $q$-beta integral in the limit $N\to\infty$. The other is a $q$-analogue of…

Ramanujan's trigonometric sum $c_q(n)$ can be interpreted as a set of trigonometric moments of a finite measure concentrated at primitive $q$-th roots of unity with equal masses. This gives rise to sets of corresponding polynomials…

We establish a new multiplicity lemma for solutions of a differential system extending Ramanujan's classical differential relations. This result can be useful in the study of arithmetic properties of values of Riemann zeta function at odd…

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…

This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…