The Ramanujan Journal

, Volume 19, Issue 2, pp 207–224 | Cite as

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

  • Aleksandar Ivić


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 3/8U=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.


Riemann zeta-function Divisor functions Power moments in short intervals Upper bounds 

Mathematics Subject Classification (2000)

11M06 11N37 


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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Katedra Matematike RGF-aUniversitet u BeograduBelgradeSerbia

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