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

2013/05/09 by Ivić, Aleksandar
#11M06 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1305.2028

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)/(2)Δ(4x) and ∫₀^T E^*(t) dt = (3)/(4)πT + 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 formulas for integrals of the form ∫_T^2T(∫_t-H^t+H|ζ(1/2+iu)|² du)^k dt (k∈ N, 1 ≪ H ≤ T) are also treated.

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