Irrationality of the ζ Function on Odd Integers
نویسندگان
چکیده
The ζ function is defined by ζ(s) = ∑ n 1/n . This talk is a study of the irrationality of the zeta function on odd integer values > 2.
منابع مشابه
Irrationality of values of the Riemann zeta function
The paper deals with a generalization of Rivoal’s construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function ζ(s) only at odd points. We prove theorems on the irrationality of the number ζ(s) for some odd integers s in a given segment of the set of positive integers. Using certain refined arithmetical estimates, we strengthen Rivoal’s origin...
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The story exposed in this paper starts in 1978, when R. Apéry [Ap] gave a surprising sequence of exercises demonstrating the irrationality of ζ(2) and ζ(3). (For a nice explanation of Apéry’s discovery we refer to the review [Po].) Although the irrationality of the even zeta values ζ(2), ζ(4), . . . for that moment was a classical result (due to L. Euler and F. Lindemann), Apéry’s proof allows ...
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A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of ζ(2) and ζ(3), as well as to explain Rivoal’s recent result (math.NT/0008051) on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers ζ(5), ζ(7),...
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