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

Hideaki Ishikawa; Kohji Matsumoto

Displaying similar documents to “An explicit formula of Atkinson type for the product of the Riemann zeta-function and a Dirichlet polynomial”

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 $1 / 2 < σ < 1$ .

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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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

On zeta-functions associated to certain cusp forms. II

Antanas Laurinčikas, Joern Steuding, Darius Šiaučiūnas (2009)

Open Mathematics

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A formula for the mean value of multiplicative functions associated to certain cusp forms is obtained. The paper is a continuation of [4].

An integral involving the generalized zeta function.

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

International Journal of Mathematics and Mathematical Sciences

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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.

Upper bounds for the moments of derivatives of Dirichlet L-functions

Keiju Sono (2014)

Open Mathematics

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In this paper, we give certain upper bounds for the 2k-th moments, k ≥ 1/2, of derivatives of Dirichlet L-functions at s = 1/2 under the assumption of the Generalized Riemann Hypothesis.

On differences of two squares

Manfred Kühleitner, Werner Nowak (2006)

Open Mathematics

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The arithmetic function ρ(n) counts the number of ways to write a positive integer n as a difference of two squares. Its average size is described by the Dirichlet summatory function Σn≤x ρ(n), and in particular by the error term R(x) in the corresponding asymptotics. This article provides a sharp lower bound as well as two mean-square results for R(x), which illustrates the close connection between ρ(n) and the number-of-divisors function d(n).

On asymptotic constants related to products of Bernoulli numbers and factorials.

Kellner, Bernd C. (2009)

Integers

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Note on a Paper by h. l. Montgomery ``omega Theorems for the Riemann Zeta-function''

K. Ramachandra, A. Sankaranarayanan (1991)

Publications de l'Institut Mathématique

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Note on a Paper by h. l. Montgomery ``omega Theorems for the Riemann Zeta-function''

K. Ramachandra, A. Sankaranarayanan (1991)

Publications de l'Institut Mathématique

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On the mean value of the generalized Dirichlet $L$ -functions

Rong Ma, Yuan Yi, Yulong Zhang (2010)

Czechoslovak Mathematical Journal

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Let $q \geq 3$ be an integer, let $χ$ denote a Dirichlet character modulo $q .$ For any real number $a \geq 0$ we define the generalized Dirichlet $L$ -functions $L (s, χ, a) = \sum_{n = 1}^{\infty} \frac{χ (n)}{{(n + a)}^{s}},$ where $s = σ + i t$ with $σ > 1$ and $t$ both real. They can be extended to all $s$ by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet $L$ -functions especially for $s = 1$ and $s = \frac{1}{2} + i t$ , and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.