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In this paper, we establish a lower bound for the dimension of the vector spaces spanned over ? by 1 and the sums of the values of the Riemann zeta function at even and odd points. As a consequence, we obtain numerical results on the irrationality and linear independence of the sums of zeta values at even and odd points from a given interval of the positive integers. 相似文献
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Andrey Bovykin 《Proceedings of the American Mathematical Society》2007,135(9):2967-2973
For every and every function of one argument, we introduce the statement : ``for all , there is such that for any set of rational numbers, there is of size such that for any two -element subsets and in , we have We prove that for and any function eventually dominated by , the principle is not provable in . In particular, the statement is not provable in Peano Arithmetic. In dimension 2, the result is: does not prove , where and is the inverse of the Ackermann function.
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J. J. Mačys 《Lithuanian Mathematical Journal》2008,48(4):401-404
An elementary proof of the irrationality of the number e is well known, while this is not the case with the number π or even,
e.g., e7. In this note, simple and quite transparent proofs of the irrationality of the number π and rational degrees of e, as well
as some other values of elementary functions, are proposed.
The research was partially supported by the Lithuanian State Science and Studies Foundation (grant No T-25/08). 相似文献
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Certain q-analogs h
p(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erds (J. Indiana Math. Soc.
12, 1948, 63–66). In 1991–1992 Peter Borwein (J. Number Theory
37, 1991, 253–259; Proc. Cambridge Philos. Soc.
112, 1992, 141–146) used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs ln
p
(2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger (Adv. Appl. Math.
20, 1998, 275–283) used the qEKHAD symbolic package to find q-WZ pairs that provide a proof of irrationality similar to Apéry's proof of irrationality of (2) and (3). They also obtain an upper bound for the measure of irrationality, but better upper bounds were earlier given by Bundschuh and Väänänen (Compositio Math.
91, 1994, 175–199) and recently also by Matala-aho and Väänänen (Bull. Australian Math. Soc.
58, 1998, 15–31) (for ln
p
(2)). In this paper we show how one can obtain rational approximants for h
p(1) and ln
p
(2) (and many other similar quantities) by Padé approximation using little q-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Väänänen for h
p(1) and a better upper bound as the one given by Matala-aho and Väänänen for ln
p
(2). 相似文献
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Mathematical Notes - 相似文献
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通过理性与非理性概念的简要比较,结合相关的建筑设计作品,分析了非理性成分在建筑设计以及城市设计中的表现形式.论证了非理性成分在建筑设计中的现实意义及发展趋势,它的进一步发展将对当代的建筑潮流的发展产生深远的影响. 相似文献
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Frits Beukers 《数学学报(英文版)》2008,24(4):663-686
We give a proof of the irrationality of p-adic zeta-values ξp(κ) for p = 2, 3 and κ = 2,3.Such results were recently obtained by Calegari as an application of overconvergent p-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show the irrationality of some other p-adic L-series values, and values of the p-adic Hurwitz zeta-function. 相似文献
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The paper gives bounds for the approximation of the values of Ramanujan's Mock Theta functions of third order and more generally of some q-hypergeometric functions by the elements of an algebraic number field. Simultaneous approximations for the values of q-exponential function are also obtained. All the results are given both in the archimedean and p-adic case. 相似文献