Euler-type Multiple Integrals as Linear Forms in Zeta Values
نویسندگان
چکیده
0. In 1978, Apéry showed the irrationality of ζ(3) = ∑∞ n=1 1 n3 by giving the approximants `n = unζ(3) − vn ∈ Qζ(3) + Q, un, dnvn ∈ Z, dn = l.c.m.(1, 2, . . . , n), with the property |`n| → ( √ 2 − 1) < 1/e as n → ∞. A similar approach was put forward to show the irrationality of ζ(2) (which is π/6, hence transcendental thanks to Lindemann) but I will concentrate on the case of ζ(3). A few months later Beukers represented Apéry’s approximants by means of Eulertype integrals, namely
منابع مشابه
Well-poised Hypergeometric Service for Diophantine Problems of Zeta Values
It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studing arithmetic properties of the values of Riemann’s zeta function. By these well-poised means we obtain: (1) a permutation group for linear forms in 1 and ζ(4) = π/90 yielding a conditional upper bound for the irrationality measure of ζ(4); (2) a second-order Apéry-like recursion...
متن کاملIntegrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler’s Constant
Let T be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over T , one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expre...
متن کاملWell-poised Hypergeometric Transformations of Euler-type Multiple Integrals
Several new multiple-integral representations are proved for well-poised hypergeometric series and integrals. The results yield, in particular, transformations of the multiple integrals that cannot be achieved by evident changes of variable. All this generalizes some classical results of Whipple and Bailey in analysis and, on the other hand, certain analytic constructions with known connection ...
متن کاملNew Changhee q-Euler numbers and polynomials associated with p-adic q-integrals
Using non-archimedean q-integrals on Zp defined in [15, 16], we define a new Changhee q-Euler polynomials and numbers which are different from those of Kim [7] and Carlitz [2]. We define generating functions of multiple q-Euler numbers and polynomials. Furthermore we construct multivariate Hurwitz type zeta function which interpolates the multivariate q-Euler numbers or polynomials at negative ...
متن کاملIterated Integrals of Modular Forms and Noncommutative Modular Symbols
The main goal of this paper is to study properties of the iterated integrals of modular forms in the upper halfplane, eventually multiplied by zs−1, along geodesics connecting two cusps. This setting generalizes simultaneously the theory of modular symbols and that of multiple zeta values. §0. Introduction and summary This paper was inspired by two sources: theory of multiple zeta values on the...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2003