We study the function \Delta_k(x):=\sum_{n\leq x} d_k(n) - \mbox{Res}_{s=1}
( \zeta^k(s) x^s/s ), where kβ₯3 is an integer, dkβ(n) is the k-fold
divisor function, and ΞΆ(s) is the Riemann zeta-function. For a large
parameter X, we show that there exist at least
X9637ββΞ΅ disjoint subintervals of [X,2X], each of
length X21ββΞ΅, such that β£Ξ3β(x)β£β«x1/3 for
all x in the subinterval. We show further that if the Lindel\"{o}f hypothesis
is true and kβ₯3, then there exist at least
Xk(kβ1)1ββΞ΅ disjoint subintervals of [X,2X], each of
length X1βk1ββΞ΅, such that β£Ξkβ(x)β£β«x21ββ2k1β for all x in the subinterval. If the Riemann
hypothesis is true, then we can improve the length of the subintervals to β«X1βk1β(logX)βk2β2. These results may be viewed as
higher-degree analogues of a theorem of Heath-Brown and Tsang, who studied the
case k=2. The first main ingredient of our proofs is a bound for the second
moment of Ξkβ(x+h)βΞkβ(x). We prove this bound using a method of
Selberg and a general lemma due to Saffari and Vaughan. The second main
ingredient is a bound for the fourth moment of Ξkβ(x), which we obtain
by combining a method of Tsang with a technique of Lester.Comment: 28 page