Title: Sign changes of the error term in the Piltz divisor problem
Abstract: For an integer $k\geq 3,$ let $\Delta_k(x):=\sum_{n\leq x} d_k(n) - \mbox{Res}_{s=1} ( \zeta^k(s) x^s/s )$, where $d_k(n)$ is the $k$-fold divisor function, and $\zeta(s)$ is the Riemann zeta-function. In the 1950's, Tong showed for all large enough $X,$ $\Delta_k(x)$ changes sign at least once in the interval $[X,X+C_kX^{1-1/k}]$ for some positive constant $C_k$. For a large parameter $X$, we show that if the Lindel\"{o}f hypothesis is true, then there exist many disjoint subintervals of $[X,2X]$, each of length $X^{1-\frac{1}{k}-\varepsilon}$, such that $\Delta_k(x)$ does not change sign in any of these subintervals. 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 a theorem of Heath-Brown and Tsang, who studied the case $k=2$. This is joint work with Siegfried Baluyot.
