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On (Not) Computing the Möbius Function Using Bounded Depth Circuits

Published online by Cambridge University Press:  24 August 2012

BEN GREEN
BEN GREEN*
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK (e-mail: b.j.green@dpmms.cam.ac.uk)
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Abstract

Any function F: {0,. . ., N − 1} → {−1,1} such that F(x) can be computed from the binary digits of x using a bounded depth circuit is orthogonal to the Möbius function μ in the sense that

\[ \frac{1}{N} \sum_{0 \leq x \leq N-1} \mu(x)F(x) → 0 \quad\text{as}~~ N → \infty. \]
The proof combines a result of Linial, Mansour and Nisan with techniques of Kátai and Harman, used in their work on finding primes with specified digits.

Keywords

Primary 11N99
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 6 , November 2012 , pp. 942 - 951
Copyright
Copyright © Cambridge University Press 2012

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