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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 ( 1 2 + i t ) | and the fourth moment of | ζ ( σ + i t ) | for 1 / 2 < σ < 1 .

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 1 2 + i t . If E * t = E t - 2 π Δ * t / 2 π with Δ * x = - Δ x + 2 Δ 2 x - 1 2 Δ 4 x , then we obtain 0 T E * t 4 d t e T 16 / 9 + ε . We also show how our method of proof yields the bound r = 1 R t r - G t r + G ς 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].

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.

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 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 3 be an integer, let χ denote a Dirichlet character modulo q . For any real number a 0 we define the generalized Dirichlet L -functions L ( s , χ , a ) = n = 1 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 = 1 2 + i t , and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.