Sharp Approximations for the Ramanujan Constant

TL;DR

This paper develops precise approximations for the Ramanujan constant using sine functions and polynomials, analyzing their properties and related special functions to improve understanding and computation.

Contribution

It introduces new sharp approximations for the Ramanujan constant and explores their monotonicity, concavity, and convexity properties, along with related special functions.

Findings

Established monotonicity, concavity, convexity of approximation functions

Derived properties of the Riemann zeta function and related sums

Provided improved bounds and approximations for the Ramanujan constant

Abstract

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 of certain combinations defined in terms of $R(a)$, $\sin(\pi a)$ and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too.