Sign changes of the error term in the Piltz divisor problem

2023/02/16 by Baluyot, Siegfred, Castillo, Cruz
#11M06 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2302.08003

Abstract

We study the function Δ_k(x):=∑_{n≤ x} d_k(n) - Res_s=1 ( ζ^k(s) x^s/s ), where k≥ 3 is an integer, d_k(n) is the k-fold divisor function, and ζ(s) is the Riemann zeta-function. For a large parameter X, we show that if the Lindelöf hypothesis is true, then there exist at least X^{(1)/(k(k-1))-ε} disjoint subintervals of [X,2X], each of length X^1-(1)/(k)-ε, such that |Δ_k(x)|≫ x^{(1)/(2)-(1)/(2k)} for all x in the subinterval. If the Riemann hypothesis is true, then we can improve the length of the subintervals to ≫ X^1-(1)/(k) (log X)^-k2-2. 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 Δ_k(x+h)-Δ_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 Δ_k(x), which we obtain by combining a method of Tsang with a technique of Lester.

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