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 ), $$\sin (\pi a)$$ sin ( π a ) and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too.

中文翻译：

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拉马努金常数的尖锐近似

在本文中，作者通过展示定义的某些组合的一些单调性、凹性和凸性特性，提出了所谓的拉马努金常数（或拉马努金 R 函数）R ( a ) 的正弦函数和多项式的近似值。 R ( a )、$$\sin (\pi a)$$ sin ( π a ) 和多项式的项。还介绍了黎曼 zeta 函数的一些性质及其相关的特殊和。