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

By Aleksandar Ivic. 2007

54 citations recorded by the source corpus.

AbstractWe obtain, for Tε≤U=U(T)≤T1/2−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 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 $|\zeta(\frac{1}{2}+\mathit{it})|$ is also given. Generalizations to some other number-theoretic error terms are discussed.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/44de11dcca3e2d74e6e8a070dcd65727943e3764