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V. V. Zudilin 《Mathematical Notes》2007,81(3-4):297-301
We prove two new series of Ramanujan type for 1/π2. 相似文献
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For each sN define the constant θs with the following properties: if an entire function g(z) of type t(g)<θs satisfies then g is a polynomial; conversely, for any δ>0 there exists an entire transcendental function g(z) satisfying the display conditin and t(g)<θs+δ. The result θ1=log2 is known due to Hardy and Pólya. We provide the upper bound θsπs/3 and improve earlier lower bounds due to Gelfond (1929) and Selberg (1941). 相似文献
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Mathematical Notes - 相似文献
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We refine (and give a new proof of) Nesterenko’s famous linear independence criterion from 1985, by making use of the fact
that some coefficients of linear forms may have large common divisors. This is a typical situation appearing in the context
of hypergeometric constructions of
\mathbbQ{\mathbb{Q}}-linear forms involving zeta values or their q-analogs. We apply our criterion to sharpen previously known results in this direction. 相似文献
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Many series for 1/?? were discovered since the appearance of S. Ramanujan??s famous paper ??Modular equations and approximation to ???? published in 1914. Almost all these series involve only real numbers. Recently, in an attempt to prove a series for 1/?? discovered by Z.-W.?Sun, the authors found that a series for 1/?? involving complex numbers is needed. In this article, we illustrate a method that would allow us to prove series of this type. 相似文献
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Wadim Zudilin 《Journal of Number Theory》2009,129(8):1848-1857
We present several supercongruences that may be viewed as p-adic analogues of Ramanujan-type series for 1/π and 1/π2, and prove three of these examples. 相似文献
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The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values
of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for
γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main
tools are Euler-type integrals and hypergeometric series.
2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15
The work of the second author is supported by an Alexander von Humboldt research fellowship
Dedication: To Leonhard Euler on his 300th birthday. 相似文献
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We propose hypergeometric constructions of simultaneous approximations to polylogarithms. These approximations suit for computing the values of polylogarithms and satisfy 4-term Apéry-like (polynomial) recursions. 相似文献