Sharp 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.

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Correspondence to Song-Liang Qiu.

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This research is supported by NSF of China (Grant No.11171307) and Zhejiang Provincial NSF of China (Grant No.LQ17A010010).

Communicated by Mourad Ismail.

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Qiu, S., Ma, X. & Huang, T. Sharp Approximations for the Ramanujan Constant. Constr Approx 51, 303–330 (2020). https://doi.org/10.1007/s00365-019-09464-3

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Keywords

  • The Ramanujan constant
  • Monotonicity
  • Convexity and concavity
  • Approximation
  • Functional inequalities
  • The Riemann zeta function

Mathematics Subject Classification

  • 11M06
  • 33B15
  • 33C05
  • 33F05