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On the riemann zeta-function and the divisor problem II

Authors:

Aleksandar Ivić

Affiliation:

(1) Katedra Matematike RGF-a, Universiteta u Beogradu, Dušina 7, 11000 Beograd, Serbia and Montenegro

Abstract:

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 $left| {zeta left( {frac{1}{2} + it} right)} right|$ . If E ^*(t)=E(t)-2πΔ^*(t/2π) with $Delta *left( x right) + 2Delta left( {2x} right) - frac{1}{2}Delta left( {4x} right)$ , then we obtain

$int_0^T {left| {E*left( t right)} right|^5 dt} ll _varepsilon T^{2 + varepsilon }$

and

$int_0^T {left| {E*left( t right)} right|^{frac{{544}}{{75}}} dt} ll _varepsilon T^{frac{{601}}{{225}} + varepsilon } .$

It is also shown how bounds for moments of | E ^*(t)| lead to bounds for moments of $left| {zeta left( {frac{1}{2} + it} right)} right|$ .

Keywords:

Dirichlet divisor problem Riemann zeta-function power moments of IE6" >

/content/p014r47412641065/11533_2006_Article_BF02479196_TeX2GIFIE6.gif" alt=" $$left| {zeta left( {frac{1}{2} + it} right)} right|$$ " align=" middle" border=" 0" > power moments of E ^*(t)

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