A short proof of a series evaluation in terms of harmonic numbers.

Panholzer, Alois; Prodinger, Helmut

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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

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