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Well-poised hypergeometric service for diophantine problems of zeta values

Wadim Zudilin — 2003

Journal de théorie des nombres de Bordeaux

It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studying 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 $ζ (4) = π^{4} / 90$ yielding a conditional upper bound for the irrationality measure of $ζ (4)$ ; (2) a second-order Apéry-like recursion for $ζ (4)$ and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family...

A new lower bound for ${∥ (3 / 2)}^{k} ∥$

Wadim Zudilin — 2007

Journal de Théorie des Nombres de Bordeaux

We prove that, for all integers $k$ exceeding some effectively computable number $K$ , the distance from ${(3 / 2)}^{k}$ to the nearest integer is greater than $0 . 5803^{k}$ .

Arithmetic of linear forms involving odd zeta values

Wadim Zudilin — 2004

Journal de Théorie des Nombres de Bordeaux

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 on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers $ζ (5)$ , $ζ (7)$ , $ζ (9)$ , and $ζ (11)$ is irrational.

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Currently displaying 1 – 6 of 6

Well-poised hypergeometric service for diophantine problems of zeta values

A new lower bound for ${∥ (3 / 2)}^{k} ∥$

Arithmetic of linear forms involving odd zeta values

Heine's basic transform and a permutation group for q-harmonic series

Computing powers of two generalizations of the logarithm.

On the non-quadraticity of values of the q-exponential function and related q-series

Advanced Search

Formula preview

Currently displaying 1 – 6 of 6

Well-poised hypergeometric service for diophantine problems of zeta values

A new lower bound for ∥ ( 3 / 2 ) k ∥

Arithmetic of linear forms involving odd zeta values

Heine's basic transform and a permutation group for q-harmonic series

Computing powers of two generalizations of the logarithm.

On the non-quadraticity of values of the q-exponential function and related q-series

A new lower bound for ${∥ (3 / 2)}^{k} ∥$