Non-real Poles and Irregularity of Distribution

2019/10/22 by Lowry-Duda, David
#Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1910.09969

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.

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• Correlations of error terms for weighted prime counting functions
• Measure Theoretic Aspects of Oscillations of Error Terms
• On the distribution of the error terms in the divisor and circle problems
• Exponential sums with coefficients of certain Dirichlet series