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 \(\sum _{n=1}^\infty \mu (n)z^n\), 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.
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Acknowledgements
The work described in Sect. 4 was carried out in 2013/2014 by Yiwang Chen, Daniel Hirsbrunner, M. Tip Phaovibul, Dylan Yang, and Tong Zhang as part of an undergraduate research project at the Illinois Geometry Lab [3]. Numerical computations for this work were carried out at the University of Illinois Campus Computing Cluster, a high performance computing platform.
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Hildebrand, A.J. (2018). Unexpected Regularities in the Behavior of Some Number-Theoretic Power Series. In: Pintz, J., Rassias, M. (eds) Irregularities in the Distribution of Prime Numbers. Springer, Cham. https://doi.org/10.1007/978-3-319-92777-0_6
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DOI: https://doi.org/10.1007/978-3-319-92777-0_6
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