On the higher moments of the error term in the divisor problem

Details On the higher moments of the error term in the divisor problem On the higher moments of the error term in the divisor problem

2004-11-24

arxiv.org/abs/math/0411537v1

Illinois Journal of Mathematics 51(2007), 353-377.

Mathematics

Number Theory

27 pages

Scientific paper

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas $$ \int_1^X \Delta^3(x){\rm d}x = BX^{7/4} + O_\epsilon(X^{\beta+\epsilon}) \qquad(B > 0) $$ and $$ \int_1^X \Delta^4(x){\rm d}x = CX^2 + O_\epsilon(X^{\gamma+\epsilon}) \qquad(C > 0) $$ with $\beta = 7/5, \gamma = 23/12$. This improves on the values $\beta = 47/28, \gamma = 45/23$, due to K.-M. Tsang. A result on the integrals of $\Delta^3(x)$ and $\Delta^4(x)$ in short intervals is also proved.

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