Estimates of convolutions of certain number-theoretic error terms.

Ivić, Aleksandar

Displaying similar documents to “Estimates of convolutions of certain number-theoretic error terms.”

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

Aleksandar Ivić (2006)

Publications de l'Institut Mathématique

Similarity:

On the riemann zeta-function and the divisor problem II

Aleksandar Ivić (2005)

Open Mathematics

Similarity:

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 $|ζ (\frac{1}{2} + i t)|$ .

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

Similarity:

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

Similarity:

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 Mean Square of the Riemann Zeta Function and the Divisor Problem

Yifan Yang (2008)

Publications de l'Institut Mathématique

Similarity:

An integral involving the generalized zeta function.

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

International Journal of Mathematics and Mathematical Sciences

Similarity:

Mean values connected with the Dedekind zeta-function of a non-normal cubic field

Guangshi Lü (2013)

Open Mathematics

Similarity:

After Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K 3/ℚ, i.e. $S_{l, K_{3}} (x) = \sum_{m ⩽ x} M^{l} (m)$ , where M(m) denotes the number of integral ideals of the field K 3 of norm m and l ∈ ℕ. We improve the previous results for $S_{2, K_{3}} (x)$ and $S_{3, K_{3}} (x)$ .

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

Similarity:

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

Aleksandar Ivić (2003)

Journal de théorie des nombres de Bordeaux

Similarity:

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.

Upper bounds for the density of universality. II

Jörn Steuding (2005)

Acta Mathematica Universitatis Ostraviensis

Similarity:

We prove explicit upper bounds for the density of universality for Dirichlet series. This complements previous results [15]. Further, we discuss the same topic in the context of discrete universality. As an application we sharpen and generalize an estimate of Reich concerning small values of Dirichlet series on arithmetic progressions in the particular case of the Riemann zeta-function.