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

الأصل · EN

We obtain, for Tε≤ U=U(T)≤ T¹/²⁻ε, asymptotic formulas for ∫ₜ²ᵗ(E(t+U) - E(t))² dt, ∫ₜ²ᵗ(Δ(t+U) - Δ(t))² dt, 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 |ζ(1/2+it)|. Upper bounds of the form Oε(T¹⁺εU²) for the above integrals with biquadrates instead of square are shown to hold for T³/⁸ ≤ U =U(T) ≪ T¹/². The connection between the moments of E(t+U) - E(t) and |ζ(1/2+it)| is also given. Generalizations to some other number-theoretic error terms are discussed.