Sign Changes of the Error Term in the Piltz Divisor Problem.

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. [ABSTRACT FROM AUTHOR]