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 Lindel\"{ o}f hypothesis 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. 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 $\Delta_k(x+h)-\Delta_k(x)$. 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.
Categories: math.NT