Some definite integrals associated with the Riemann zeta function.

Srivastava, H.M.; Glasser, M.L.; Adamchik, V.S.

Displaying similar documents to “Some definite integrals associated with the Riemann zeta function.”

An integral involving the generalized zeta function.

Elizalde, E., Romeo, A. (1990)

International Journal of Mathematics and Mathematical Sciences

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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 $0 \leq ℜ s_{j} < 1 (j = 1, 2)$ .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.

Zeta functions for the Riemann zeros

André Voros (2003)

Annales de l’institut Fourier

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A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structures, plus countably many special values) are explicitly displayed.

On the Mean Square of the Riemann Zeta Function and the Divisor Problem

Yifan Yang (2008)

Publications de l'Institut Mathématique

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Some problems on mean values of the Riemann zeta-function

Aleksandar Ivić (1996)

Journal de théorie des nombres de Bordeaux

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Several problems and results on mean values of $ζ (s)$ are discussed. These include mean values of $| ζ (\frac{1}{2} + i t) |$ and the fourth moment of $| ζ (σ + i t) |$ for $1 / 2 < σ < 1$ .

An explicit formula of Atkinson type for the product of the Riemann zeta-function and a Dirichlet polynomial

Hideaki Ishikawa, Kohji Matsumoto (2011)

Open Mathematics

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We prove an explicit formula of Atkinson type for the error term in the asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial. To deal with the case when coefficients of the Dirichlet polynomial are complex, we apply the idea of the first author in his study on mean values of Dirichlet L-functions.

On the modulus of the Riemann zeta function in the critical strip

Filip Saidak, Peter D. Zvengrowski (2003)

Mathematica Slovaca

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On the riemann zeta-function and the divisor problem II

Aleksandar Ivić (2005)

Open Mathematics

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Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of $|ζ (\frac{1}{2} + i t)|$ . If E *(t)=E(t)-2πΔ*(t/2π) with $Δ * (x) + 2 Δ (2 x) - \frac{1}{2} Δ (4 x)$ , then we obtain $\int_{0}^{T} {|E * (t)|}^{5} d t ≪_{ε} T^{2 + ε}$ and $\int_{0}^{T} {|E * (t)|}^{\frac{544}{75}} d t ≪_{ε} T^{\frac{601}{225} + ε} .$ It is also shown how bounds for moments of | E *(t)| lead to bounds for moments of $|ζ (\frac{1}{2} + i t)|$ .

Integral Representations of the Logarithmic Derivative of the Selberg Zeta Function

Gušić, Dženan (2010)

Mathematica Balkanica New Series

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AMS Subj. Classiﬁcation: MSC2010: 11F72, 11M36, 58J37 We point out the importance of the integral representations of the logarithmic derivative of the Selberg zeta function valid up to the critical line, i.e. in the region that includes the right half of the critical strip, where the Euler product deﬁnition of the Selberg zeta function does not hold. Most recent applications to the behavior of the Selberg zeta functions associated to a degenerating sequence of ﬁnite volume,...