On The Sixth-power Moment Of △(x)

Dirichlet divisor problem is a classical problem in analytiv number theory. Let d(n) be the Dirichlet divisor function and△(x) denote the error term of the sum∑n≤x d(n) for a large real variable x. Then Dirichlet proved that△(x)=O(x1/2). The exponent 1/2 was improved by many authors. The latest result reads proved by Huxley. Tong proved that holds for A0=35/4. It is conjectured that which is supported by (0.2) and (0.3).Tsang first studied the third and fourth-power moments of△(x). He proved that whereδ3=1/14,δ4=1/23, and In [6] Zhai proved that (0.5) holds forδ3=1/4. Ivic and Sargos[7] proved that (0.5) holds forδ3=7/20.Following the approach of Tsang, Zhai proved that (0.0) holds forδ4= 2/41. This approach used the method of exponential sums. In particular, if the exponent pair conjecture is true, namely, if (∈,1/2+∈) is an exponent pair, then (0.6) holds forδ4=1/14. However, in [7] Ivic and Sargos proved a substantially better result. They proved that (0.6) holds forδ4=1/12. Recently, combining the method of [7] and a recent deep result of Robert and Sargos, Zhai proved that (0.6) holds forδ4=3/28.By a unified approach, Zhai proved that the asymptotic formula holds for 3≤k≤9, where Ck and 0<δk<1 are explicit constants. The asymp-totic formula (0.7) improved the result of Heath-Brown . Especially when k=5, the asymptotic formula (0.7) holds forδ5=1/64, which improved an earlier expo-nentδ5=5/816 proved in [6] by the approach of Tsang.Zhang and Zhai proved that where Which improved the result of zhai.The aim of this paper is to study the sixth power moment of△(x). Our main result is the following Theorem.For fixed T≥10,we have where注.Our theorem improves the value ofδ6=1/64,due to zhai.