Sign changes of the error term in the Piltz divisor problem

TL;DR

This paper investigates the sign changes of the error term in the Piltz divisor problem for higher degrees, establishing lower bounds on the number of intervals where the error term is significantly large, under the assumptions of the Lindel"{o}f} hypothesis and the Riemann hypothesis.

Contribution

It provides new bounds on the frequency and size of sign changes of the error term in the Piltz divisor problem for general k, extending previous results for lower degrees.

Findings

Under Lindel"{o}f} hypothesis, many disjoint subintervals with large error term exist.

Under Riemann hypothesis, the subintervals where the error is large can be made longer.

The paper develops bounds for moments of the error term using advanced analytic techniques.

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 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.