The Fourier expansion of Epstein's zeta function over an algebraic number field and its consequences for algebraic number theory

A. Terras

Displaying similar documents to “The Fourier expansion of Epstein's zeta function over an algebraic number field and its consequences for algebraic number theory”

The Fourier expansion of Epstein's zeta function for totally real algebraic number fields and some consequences for Dedekind's zeta function

A. Terras (1976)

Acta Arithmetica

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On Epstein's zeta function

Paul Bateman, E. Grosswald (1964)

Acta Arithmetica

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Enumerating non-equivalent matrices over principal ideal domains

John Knopfmacher (1994)

Mathematica Slovaca

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Zeta functions of quadratic forms

K. Ramanathan (1961)

Acta Arithmetica

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An application of Mellin-Barnes' type integrals to the mean square of Lerch zeta-functions.

Masanori Katsurada (1997)

Collectanea Mathematica

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We shall establish full asymptotic expansions for the mean squares of Lerch zeta-functions, based on F. V. Atkinson's device. Mellin-Barnes' type integral expression for an infinite double sum will play a central role in the derivation of our main formulae.

On Ramanujan's formula for values of Riemann zeta-function at positive odd integers

Koji Katayama (1973)

Acta Arithmetica

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Some formulas for the Riemann zeta function at odd integer argument resulting from Fourier expansions of the Epstein zeta function

A. Terras (1976)

Acta Arithmetica

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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 $k \geq 3$ , and fixed $\frac{1}{2} < σ < 1$ we consider $\int_{1}^{T} {|ζ (σ + i t)|}^{2 k} d t = \sum_{n = 1}^{\infty} d_{k}^{2} (n) n^{- 2 σ} T + R (k, σ; T),$ where $R (k, σ; T) = 0 (T) (T \to \infty)$ 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.