Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T05:05:03.347Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Allender, E., Saks, M. and Shparlinski, I. (2001) A lower bound for primality. J. Comput. System Sci. (Special Issue on the Fourteenth Annual IEEE Conference on Computational Complexity) 62 356–366.Google Scholar
[2]Baker, R. C. and Harman, G. (1991) Exponential sums formed with the Möbius function. J. London Math. Soc. (2) 43 193–198.Google Scholar
[3]Bernasconi, A., Damm, C. and Shparlinski, I. (1999) On the average sensitivity of testing square-free numbers. In Computing and Combinatorics: Tokyo 1999, Vol. 1627 of Lecture Notes in Computer Science, Springer, pp. 291–299.Google Scholar
[4]Davenport, H. (1937) On some infinite series involving arithmetical functions II. Quart. J. Math. Oxford 8 313–320.Google Scholar
[5]Harman, G. and Kátai, I. (2008) Primes with preassigned digits II. Acta Arith. 133 171–184.Google Scholar
[6]Iwaniec, H. and Kowalski, E. (2004) Analytic Number Theory, Vol. 53 of American Mathematical Society Colloquium Publications, AMS.Google Scholar
[7]Kalai, G. (2011) The AC0 prime number conjecture. Blog post, available at: http://gilkalai.wordpress.com/2011/02/21/the-ac0-prime-number-conjecture/Google Scholar
[8]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-functionsGoogle Scholar
[9]Kátai, I. (1986) Distribution of digits of primes in q-ary canonical form. Acta Math. Hungar. 47 341–359.Google Scholar
[10]Linial, N., Mansour, Y. and Nisan, N. (1993) Constant depth circuits, Fourier transform, and learnability. J. Assoc. Comput. Mach. 40 607–620.Google Scholar
[11]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/Google Scholar
[12]Mauduit, C. and Rivat, J. (2010) Sur un probléme de Gelfond: La somme des chiffres des nombres premiers. Ann. of Math. (2) 171 1591–1646.Google Scholar
[13]Montgomery, H. and Vaughan, R. C. (2007) Multiplicative Number Theory I: Classical Theory, Vol. 97 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
[14]Ramaré, O. and Rumely, R. (1996) Primes in arithmetic progressions. Math. Comp. 65 397–425.Google Scholar
[15]Sarnak, P. (2010) Möbius Randomness and Dynamics. Lecture notes, available at: http://www.math.princeton.edu/sarnak/MobiuslecturesSummer2010.pdfGoogle Scholar