Sign Changes of the Error Term in the Piltz Divisor Problem

Abstract 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 $k\geq 3$ 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 (LH) is true, then there exist at least $X^{\frac{1}{k(k-1)}-\varepsilon }$ disjoint subintervals of $[X,2X]$, each of length $X^{1-\frac{1}{k}-\varepsilon }$, such that $|\Delta _{k}(x)|\gg x^{\frac{1}{2}-\frac{1}{2k}}$ for all $x$ in the subinterval. In particular, $\Delta _{k}(x)$ 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 $\gg X^{1-\frac{1}{k}} (\log X)^{-k^{2}-2}$. 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 $\Delta _{k}(x+h)-\Delta _{k}(x)$. 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 $\Delta _{k}(x)$, which we obtain by combining a method of Tsang with a technique of Lester.