On some mean value results for the zeta-function in short intervals

Aleksandar Ivić

Acta Arithmetica, Tome 166 (2014), p. 141-158 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $Δ (x)$ denote the error term in the Dirichlet divisor problem, and let E(T) denote the error term in the asymptotic formula for the mean square of |ζ(1/2+it)|. If E*(t) := E(t) - 2πΔ*(t/(2π)) with Δ*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x) and $\int_{0}^{T} E * (t) d t = 3 / 4 π T + R (T)$ , then we obtain a number of results involving the moments of |ζ(1/2+it)| in short intervals, by connecting them to the moments of E*(T) and R(T) in short intervals. Upper bounds and asymptotic formulae for integrals of the form ∫T2T(∫t-Ht+H |ζ(1/2+iu|2 duk dt $(k \in ℕ, 1 ≪ H \leq T)$ are also treated.

Publié le : 2014-01-01

Zbl 06256102

EUDML-ID : urn:eudml:doc:279177

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     title = {On some mean value results for the zeta-function in short intervals},
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     volume = {166},
     year = {2014},
     pages = {141-158},
     zbl = {06256102},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa162-2-2}
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Aleksandar Ivić. On some mean value results for the zeta-function in short intervals. Acta Arithmetica, Tome 166 (2014) pp. 141-158. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa162-2-2/