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Linear independence of linear forms in polylogarithms
For x ∈ C, |x | < 1, s ∈ N, let Lis(x) be the s-th polylogarithm of x . We prove that for any non-zero algebraic number α such that |α| < 1, the Q(α)-vector space spanned by 1, Li1(α), Li2(α), . . .Expand
IRRATIONALITY AND NONQUADRATICITY MEASURES FOR LOGARITHMS OF ALGEBRAIC NUMBERS
Abstract Let 𝕂⊂ℂ be a number field. We show how to compute 𝕂-irrationality measures of a number ξ∉𝕂, and 𝕂-nonquadraticity measures of ξ if [𝕂(ξ):𝕂]>2. By applying the saddle point method to aExpand
Hypergeometric rational approximations to ζ(4)
Abstract We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novelExpand
On hypergeometric identities related to zeta values
Two linear forms, $$\sigma _n \zeta (5)+\tau _n \zeta (3)+\varphi _n$$σnζ(5)+τnζ(3)+φn and $$\sigma _n\zeta (2)+\tau _n/2$$σnζ(2)+τn/2, with suitable rational coefficients $$\sigma _n,\tau _n,\varphiExpand
Some remarks in elementary prime number theory
We give an outline of a generalization of the Gelfond-Schnirelmann method in elementary number theory. It is related to an integral of Selberg (1944) generalizing the Euler beta integral. The resultExpand
Simultaneous Approximations to ζ(2) and ζ(4)
A classical functional generalization of the first Barnes lemma
We give a brief account and a simpler proof of a contour integral formula for the Gauss hypergeometric function. Such formula is alternative to Barnes's integral formula and generalizes the firstExpand
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