Displaying similar documents to “Euler's formulae for ζ ( 2 n ) and products of Cauchy variables.”

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.

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

Large dimensional sets not containing a given angle

Viktor Harangi (2011)

Open Mathematics

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We say that a set in a Euclidean space does not contain an angle α if the angle determined by any three points of the set is not equal to α. The goal of this paper is to construct compact sets of large Hausdorff dimension that do not contain a given angle α ∈ (0,π). We will construct such sets in ℝn of Hausdorff dimension c(α)n with a positive c(α) depending only on α provided that α is different from π/3, π/2 and 2π/3. This improves on an earlier construction (due to several authors)...