Displaying similar documents to “On the riemann zeta-function and the divisor problem”

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 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 1 2 + i t . If E *(t)=E(t)-2πΔ*(t/2π) with Δ * x + 2 Δ 2 x - 1 2 Δ 4 x , then we obtain ∫ 0 T E * t 5 d t ≪ ε T 2 + ε and ∫ 0 T E * t 544 75 d t ≪ ε T 601 225 + ε . It is also shown how bounds for moments of | E *(t)| lead to bounds for moments of ζ 1 2 + i t .

On some problems involving Hardy’s function

Aleksandar Ivić (2010)

Open Mathematics

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Some problems involving the classical Hardy function Z t = ζ 1 2 + i t χ 1 2 + i t - 1 1 2 2 , ζ s = χ s ζ 1 - s , are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.

Some subclasses of close-to-convex functions

Adam Lecko (1993)

Annales Polonici Mathematici

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For α ∈ [0,1] and β ∈ (-π/2,π/2) we introduce the classes C β ( α ) defined as follows: a function f regular in U = z: |z| < 1 of the form f ( z ) = z + ∑ n = 1 ∞ a n z n , z ∈ U, belongs to the class C β ( α ) if R e e i β ( 1 - α ² z ² ) f ' ( z ) < 0 for z ∈ U. Estimates of the coefficients, distortion theorems and other properties of functions in C β ( α ) are examined.