Displaying similar documents to “An Alternative Form of the Functional Equation for Riemann’s Zeta Function, II”

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 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.

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

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.