Sign changes of the error term in the Piltz divisor problem

Speaker: Cruz Castillo

Date: Mon, Sep 25, 2023

Location: PIMS, University of Lethbridge, Online

Conference: Analytic Aspects of L-functions and Applications to Number Theory

Subject: Mathematics, Number Theory

Class: Scientific

CRG: L-Functions in Analytic Number Theory


For an integer k≥3; Δk (x) :=∑n≤xdk(n)-Ress=1 (ζk(s)xs/s), where dk(n) is the k-fold divisor function, and ζ(s) is the Riemann zeta-function. In the 1950's, Tong showed for all large enough X; Δk(x) changes sign at least once in the interval [X, X + CkX1-1/k] for some positive constant Ck. For a large parameter X, we show that if the Lindelöf hypothesis is true, then there exist many disjoint subintervals of [X, 2X], each of length X1-1/k-ε such that Δk (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 << X1-1/k (logX)-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 Siegfred Baluyot.