Allowing sign-changing coefficients in Landau’s method

Develop a modification of Landau’s method, as used in the analysis of sharp cubic-field counting, that permits partial-sum estimates for zeta functions whose coefficients are not necessarily nonnegative, so as to exploit cancellation between coefficients of different Shintani zeta functions and improve the resulting error term.

Background

The sharp counting argument applies a result requiring the coefficients of the relevant Shintani zeta functions to be nonnegative. This requirement prevents the authors from using cancellation between different zeta functions in the sharp-counting problem, even though such cancellation is available in the smoothed problem.

The unresolved methodological problem is to extend Landau’s method to sign-changing coefficients. Such an extension would likely improve the error term for the sharp counting function, although the paper notes that obtaining a square-root error term would still appear difficult even under the Generalised Riemann Hypothesis.