Non-real Poles and Irregularity of Distribution

Abstract

We study the general theory of weighted Dirichlet series and associated summatory functions of their coefficients. We show that any non-real pole leads to oscillatory error terms. This applies even if there are infinitely many non-real poles with the same real part. Further, we consider the case when the non-real poles lie near, but not on, a line. The method of proof is a generalization of classical ideas applied to study the oscillatory behavior of the error term in the prime number theorem.

7 pages (with minor correction)

References in corpus (1)

• A note on the Laplace transform of the square in the circle problem

Cited by in corpus (1)

• Triple Correlation Sums of Coefficients of Cusp Forms