On the Riemann zeta-function and the divisor problem IV

Details On the Riemann zeta-function and the divisor problem IV On the Riemann zeta-function and the divisor problem IV

2007-01-07

arxiv.org/abs/math/0701202v2

Uniform Distribution Theory 1(2006), 125-135

Mathematics

Number Theory

11 pages

Scientific paper

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - {1\over2}\Delta(4x)$, then it is proved that $$ \int_0^T|E^*(t)|^3dt \ll_\epsilon T^{3/2+\epsilon}, $$ which is (up to `$\epsilon$' best possible) and $\zeta(1/2+it) \ll_\epsilon t^{\rho/2+\epsilon}$ if $E^*(t) \ll_\epsilon t^{\rho+\epsilon}$.

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