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On the divisor function and the Riemann zeta-function in short intervals

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Abstract

We obtain, for T ε≤U=U(T)≤T 1/2−ε, asymptotic formulas for

$$\int_{T}^{2T}\Bigl(E(t+U)-E(t)\Bigr)^{2}{\mathrm{d}}{t},\qquad \int_{T}^{2T}\Bigl(\Delta (t+U)-\Delta (t)\Bigr)^{2}{\mathrm{d}}{t},$$

where Δ(x) is the error term in the classical divisor problem, and E(T) is the error term in the mean square formula for \(|\zeta(\frac{1}{2}+\mathit{it})|\) . Upper bounds of the form O ε (T 1+ε U 2) for the above integrals with biquadrates instead of square are shown to hold for T 3/8≤U=U(T)≪ T 1/2. The connection between the moments of E(t+U)−E(t) and \(|\zeta(\frac{1}{2}+\mathit{it})|\) is also given. Generalizations to some other number-theoretic error terms are discussed.

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  1. Katedra Matematike RGF-a, Universitet u Beogradu, Ðušina 7, 11000, Belgrade, Serbia

    Aleksandar Ivić

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Ivić, A. On the divisor function and the Riemann zeta-function in short intervals. Ramanujan J 19, 207–224 (2009). https://doi.org/10.1007/s11139-008-9142-0

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