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

In: Irregularities in the Distribution of Prime Numbers

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  • A. J. Hildebrand

    (University of Illinois, Department of Mathematics)

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.

Suggested Citation

  • A. J. Hildebrand, 2018. "Unexpected Regularities in the Behavior of Some Number-Theoretic Power Series," Springer Books, in: János Pintz & Michael Th. Rassias (ed.), Irregularities in the Distribution of Prime Numbers, pages 111-123, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-92777-0_6
    DOI: 10.1007/978-3-319-92777-0_6
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