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

Aleksandar Ivić. "On some mean value results for the zeta-function in short intervals." Acta Arithmetica 162.2 (2014): 141-158. <http://eudml.org/doc/279177>.

@article{AleksandarIvić2014,
abstract = {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 $∫_0^T E*(t) dt = 3/4πT + R(T)$, 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 ∈ ℕ, 1 ≪ H ≤ T) $are also treated. },
author = {Aleksandar Ivić},
journal = {Acta Arithmetica},
keywords = {Dirichlet divisor problem; Riemann zeta-function; integral of the error term; mean value estimates; short intervals},
language = {eng},
number = {2},
pages = {141-158},
title = {On some mean value results for the zeta-function in short intervals},
url = {http://eudml.org/doc/279177},
volume = {162},
year = {2014},
}

TY - JOUR
AU - Aleksandar Ivić
TI - On some mean value results for the zeta-function in short intervals
JO - Acta Arithmetica
PY - 2014
VL - 162
IS - 2
SP - 141
EP - 158
AB - 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 $∫_0^T E*(t) dt = 3/4πT + R(T)$, 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 ∈ ℕ, 1 ≪ H ≤ T) $are also treated.
LA - eng
KW - Dirichlet divisor problem; Riemann zeta-function; integral of the error term; mean value estimates; short intervals
UR - http://eudml.org/doc/279177
ER -