On the riemann zeta-function and the divisor problem II

Aleksandar Ivić

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On the riemann zeta-function and the divisor problem

Aleksandar Ivić (2004)

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 $|ς (\frac{1}{2} + i t)|$ . If $E^{*} (t) = E (t) - 2 π Δ^{*} (t / 2 π)$ with $Δ^{*} (x) = - Δ (x) + 2 Δ (2 x) - \frac{1}{2} Δ (4 x)$ , then we obtain $\int_{0}^{T} {(E^{*} (t))}^{4} d t ≪_{e} T^{16 / 9 + ε}$ . We also show how our method of proof yields the bound $\sum_{r = 1}^{R} {(\int_{t r - G}^{t r + G} {|ς (\frac{1}{2} + i t)|}^{2} d t)}^{4} ≪_{e} T^{2 + e} G^{- 2} + R G^{4} T^{ε}$ , where T 1/5+ε≤G≪T, T

An integral involving the generalized zeta function.

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

International Journal of Mathematics and Mathematical Sciences

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Estimates of convolutions of certain number-theoretic error terms.

Ivić, Aleksandar (2004)

International Journal of Mathematics and Mathematical Sciences

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On the Selberg Integral of the $k$ -divisor Function and the $2 k$ -th Moment of the Riemann Zeta-function

Giovanni Coppola (2010)

Publications de l'Institut Mathématique

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Convolutions and Mean Square Estimates of Certain Number-theoretic Error Terms

Aleksandar Ivić (2006)

Publications de l'Institut Mathématique

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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 $1 / 2 < σ < 1$ .

Some definite integrals associated with the Riemann zeta function.

Srivastava, H.M., Glasser, M.L., Adamchik, V.S. (2000)

Zeitschrift für Analysis und ihre Anwendungen

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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 a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

Manfred Kühleitner, Werner Nowak (2013)

Open Mathematics

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The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.

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.