Sign changes of the error term in the Piltz divisor problem

We study the function ${\mathrm{\Delta }}_k(x):={\sum }_{n\le x}{d}_k(n)-Re{s}_{s=1}({\zeta }^k(s)x^s\mathrm{/}s)$, where $k\ge 3$ is an integer, $d_k(n)$ is the $k$-fold divisor function, and $\zeta (s)$ is the Riemann zeta-function. For a large parameter $X$, we show that if the Lindelöf hypothesis is true, then there exist at least $X^{\frac{1}{k(k-1)}-\epsilon }$ disjoint subintervals of $[X,2X]$, each of length $X^{1-\frac{1}{k}-\epsilon }$, such that $\mathrm{\mid }{\mathrm{\Delta }}_k(x)\mathrm{\mid }\gg x^{\frac{1}{2}-\frac{1}{2k}}$ for all $x$ in the subinterval. If the Riemann hypothesis is true, then we can improve the length of the subintervals to $\gg X^{1-\frac{1}{k}}(\log X{)}^{-k^2-2}$. 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 ${\mathrm{\Delta }}_k(x+h)-{\mathrm{\Delta }}_k(x)$. 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 ${\mathrm{\Delta }}_k(x)$, which we obtain by combining a method of Tsang with a technique of Lester.