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Inequalities for the Landau constant. (English) Zbl 0984.11068
The authors prove an asymptotic expansion for the \(n\)th Landau constant \[ G_n=\sum _{i=0}^n \frac 1{2^{4i}} \binom{2i}{i}. \] This leads to a sharp inequality \[ 1.0663<G_n-\frac 1\pi \psi \left(n+\frac 5 4\right)<1.0724, \] where \(\psi(x)\) is the logarithmic derivative of the \(\Gamma \)-function.

MSC:
11Y60 Evaluation of number-theoretic constants
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References:
[1] Handbook of Mathematical Functions with Formulas. Graphs and Mathematical Tables (M. Abramowitz, I. Stegun, Dover Publications Inc., New York, 1972. · Zbl 0543.33001
[2] BRUTMAN L.: A sharp estimate of the Landau constants. J. Approx. Theory 34 (1982), 217-220. · Zbl 0486.41013
[3] FALALEEV L. P.: Inequalities for the Landau constants. Siberian Math. J. 32 (1991), 896-897. · Zbl 0778.41014
[4] GRADSHTEYN I. S.-RYZHIK I. M.: Table of Integrals, Series, and Products. Academic Press, New York-London-Toronto, 1980. · Zbl 0521.33001
[5] GURLAND J.: On Wallis’s formula. Amer. Math. Monthly 63 (1956), 643-645. · Zbl 0073.14601
[6] HANSEN E. R.: A Table of Series and Products. Prentice-Hall, Englewood Cliffs, N.J., 1975. · Zbl 0438.00001
[7] LANDAU E.: Abschtzung der Koeffiziententensumme einer Potenzreihe. Arch. Math. Phys. 21 (1913), 42-50.
[8] PALAGALLO J. A.-PRICE, Jr. T. E.: Near-best approximation by averaging polynomial interpolants. IMA J. Numer. Anal. 7 (1987), 107-122. · Zbl 0621.41005
[9] WATSON G. N.: The constants of the Landau and Lebesgue. Quart. J. Math. Oxford Ser. (2) 1 (1930), 310-318. · JFM 56.0972.02
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