Sign Changes of the Error Term in the Piltz Divisor Problem

Abstract We study the function ₊ (x): = ₍ ₗ d₊ (n) - Resₒ=₁ (^k (s) x^s/s), where k 3 is an integer, d₊ (n) is the k-fold divisor function, and (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^1{k (k-1) - } disjoint subintervals of X, 2X, each of length X^1-1{k- }, such that | ₊ (x) | x^1{2-12k} for all x in the subinterval. In particular, ₊ (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 X^1-1{k} (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 ₊ (x+h) - ₊ (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 ₊ (x), which we obtain by combining a method of Tsang with a technique of Lester.