Displaying similar documents to “Arithmetic of linear forms involving odd zeta values”

Well-poised hypergeometric service for diophantine problems of zeta values

Wadim Zudilin (2003)

Journal de théorie des nombres de Bordeaux

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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...

On mean values of some zeta-functions in the critical strip

Aleksandar Ivić (2003)

Journal de théorie des nombres de Bordeaux

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For a fixed integer k ≥ 3 , and fixed 1 2 < σ < 1 we consider ∫ 1 T ζ ( σ + i t ) 2 k d t = ∑ n = 1 ∞ d k 2 ( n ) n - 2 σ T + R ( k , σ ; T ) , where R ( k , σ ; T ) = 0 ( T ) ( T → ∞ ) is the error term in the above asymptotic formula. Hitherto the sharpest bounds for R ( k , σ ; T ) are derived in the range min ( β k , σ k * ) < σ < 1 . We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.

The multiple gamma function and its q-analogue

Kimio Ueno, Michitomo Nishizawa (1997)

Banach Center Publications

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We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vignéras multiple gamma function by considering the classical limit of the multiple q-gamma function.