Sign Changes of the Error Term in the Piltz Divisor Problem

We study the function Δ ₖ(x):=∑ n≤ x dₖ(n) - Resₛ₌₁ ( ζ ᵏ(s) xˢ/s ), where k≥ 3 is an integer, dₖ(n) is the k-fold divisor function, and ζ (s) is the Riemann zeta-function. For a large parameter X, we show that if the Lindelöf hypothesis (LH) is true, then there exist at least X1/k(k-1)-ε disjoint subintervals of $[X,2X]$, each of length X1-1/k-ε, such that |Δ ₖ(x)| x1/2-1/2k for all x in the subinterval. In particular, Δ ₖ(x) does not change sign in any of these subintervals. If the Riemann hypothesis (RH) is true, then we can improve the length of the subintervals to X1-1/k (log X)^-k²-2. These results may be viewed as higher-degree analogues of theorems of Heath-Brown and Tsang, who studied the case $k=2$, and Cao, Tanigawa, and Zhai, who studied the case $k=3$. The first main ingredient of our proofs is a bound for the second moment of Δ ₖ(x+h)-Δ ₖ(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 Δ ₖ(x), which we obtain by combining a method of Tsang with a technique of Lester.