On the Riemann zeta-function and the divisor problem
| dc.creator | Ivić, Aleksandar | |
| dc.date | 2004-04-14 | |
| dc.date | 2004-07-02 | |
| dc.date.accessioned | 2026-07-07T05:07:26Z | |
| dc.date.available | 2026-07-07T05:07:26Z | |
| dc.description | Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|ζ(1/2 + it)|$. If $E^*(t) = E(t) - 2πΔ^*(t/(2π))$ with $Δ^*(x) = - Δ(x) +2Δ(2x)- {1\over2}Δ(4x)$, then we obtain $$ \int_0^T(E^*(t))^4 dt \ll_εT^{16/19+varepsilon}$$, which is the first non-trivial bound for higher moments of $E^*(t)$. The method of proof also provides an upper bound for sums of fourth powers of mean square integrals of $|ζ(1/2 + it)|$ over well-spaced points. This, in turn, yields a new proof of the twelfth moment estimate for $|ζ(1/2 + it)|$. Among the chief ingredients in the proof is a recent result of Robert--Sargos on the distribution of four square roots of integers, plus an approach of M. Jutila that involves the use of Airy integrals to deal with the ensuing exponential sums. | |
| dc.description | 18 pages | |
| dc.identifier | https://arxiv.org/abs/math/0404261 | |
| dc.identifier | http://arxiv.org/abs/math/0404261 | |
| dc.identifier | Central European J. Math. 2(4) (2004), 1-15. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70859 | |
| dc.subject | Number Theory | |
| dc.subject | 11N37, 11M06 | |
| dc.title | On the Riemann zeta-function and the divisor problem | |
| dc.type | text |