Unexpected Regularities in the Behavior of Some Number-Theoretic Power Series

A. J. Hildebrand ()
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A. J. Hildebrand: University of Illinois, Department of Mathematics

A chapter in Irregularities in the Distribution of Prime Numbers, 2018, pp 111-123 from Springer

Abstract: Abstract The goal of this paper is to draw attention to a surprising and little-known phenomenon, namely the unexpected regularity in the behavior of the Möbius power series ∑ n = 1 ∞ μ ( n ) z n $$\sum _{n=1}^\infty \mu (n)z^n$$ , and some related series. This phenomenon was first pointed out and investigated a half century ago in a remarkable, but now nearly forgotten, paper by Carl-Erik Fröberg. Its manifestations include “fake” asymptotics as z → 1, and error terms that are significantly better than the usual error terms in prime number estimates. We describe these results and some recent developments, explain the underlying phenomenon, and comment on possible applications.

Keywords: Power Series; Usual Error Term; Zeta Zeros; Abel Summability; Liouville Function (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-92777-0_6

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DOI: 10.1007/978-3-319-92777-0_6

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