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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science
Reference15 articles.
1. Lipton R. J. (2011) The depth of the Möbius function. Blog post, available at: http://rjlipton.wordpress.com/2011/02/23/the-depth-of-the-mobius-function/
2. Primes in arithmetic progressions
3. Kalai G. (2011) Walsh Fourier transform of the Möbius function. Math Overflow question, available at: http://mathoverflow.net/questions/57543/walsh-fourier-transform-of-mobius-functions
4. ON SOME INFINITE SERIES INVOLVING ARITHMETICAL FUNCTIONS (II)
5. On the Average Sensitivity of Testing Square-Free Numbers
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