An Omega theorem on differences of two squares.

Kühleitner, M.

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On the riemann zeta-function and the divisor problem

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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

Approximation of $π (x)$ by $Ψ (x)$ .

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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 $f (z) = z + \sum_{n = 1}^{\infty} a_{n} z^{n}$ , 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.