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On the riemann zeta-function and the divisor problem

Authors:

Aleksandar Ivić

Affiliation:

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

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 $left| {varsigma left( {tfrac{1}{2} + it} right)} right|$ . If $E^* left( t right) = Eleft( t right) - 2pi Delta ^* left( {t / 2pi } right)$ with $Delta ^* left( x right) = - Delta left( x right) + 2Delta left( {2x} right) - tfrac{1}{2}Delta left( {4x} right)$ , then we obtain

$int_0^T {left( {E^* left( t right)} right)^4 dt ll _e T^{16/9 + varepsilon } }$

. We also show how our method of proof yields the bound

$sumlimits_{r = 1}^R {left( {int_{tr - G}^{tr + G} {left| {varsigma left( {tfrac{1}{2} + it} right)} right|^2 dt} } right)^4 ll _e T^{2 + e} G^{ - 2} + RG^4 T^varepsilon }$

, where T ^1/5+ε≤G≪T, T<t ₁<...<t _R≤2T, t _r +1−t _r≥5G (r=1, ..., R−1).

Keywords:

Dirichlet divisor problem Riemann zeta-function mean square and twelfth moment of |ξ (1/2+it)| mean fourth power of E ^*(t)

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