Displaying similar documents to “Well-poised hypergeometric service for diophantine problems of zeta values”

Arithmetic of linear forms involving odd zeta values

Wadim Zudilin (2004)

Journal de Théorie des Nombres de Bordeaux

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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 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.

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.

New integral representations for the square of the Riemann zeta-function

Andreas Guthmann (1997)

Acta Arithmetica

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Introduction. The recent discovery of an analogue of the Riemann-Siegel integral formula for Dirichlet series associated with cusp forms [2] naturally raises the question whether similar formulas might exist for other types of zeta functions. The proof of these formulas depends on the functional equation for the underlying Dirichlet series. In both cases, for ζ(s) and for the cusp form zeta functions, only a simple gamma factor is involved. The next simplest case arises when two such...

Bounds for double zeta-functions

Isao Kiuchi, Yoshio Tanigawa (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In this paper we shall derive the order of magnitude for the double zeta-functionof Euler-Zagier type in the region 0 ≤ ℜ s j < 1 ( j = 1 , 2 ) .First we prepare the Euler-Maclaurinsummation formula in a suitable form for our purpose, and then we apply the theory of doubleexponential sums of van der Corput’s type.

Integral identities and constructions of approximations to zeta-values

Yuri V. Nesterenko (2003)

Journal de théorie des nombres de Bordeaux

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Some general construction of linear forms with rational coefficients in values of Riemann zeta-function at integer points is presented. These linear forms are expressed in terms of complex integrals of Barnes type that allows to estimate them. Some identity connecting these integrals and multiple integrals on the real unit cube is proved.