On the riemann zeta-function and the divisor problem

Aleksandar Ivić

Open Mathematics, Tome 2 (2004), p. 494-508 / Harvested from The Polish Digital Mathematics Library

Résumé

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

Publié le : 2004-01-01

Zbl 1116.11067

EUDML-ID : urn:eudml:doc:268928

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     title = {On the riemann zeta-function and the divisor problem},
     journal = {Open Mathematics},
     volume = {2},
     year = {2004},
     pages = {494-508},
     zbl = {1116.11067},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02475958}
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Aleksandar Ivić. On the riemann zeta-function and the divisor problem. Open Mathematics, Tome 2 (2004) pp. 494-508. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02475958/

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