On (not) computing the Mobius function using bounded depth circuits

TL;DR

The paper investigates the limitations of bounded depth circuits in computing the Mobius function, showing orthogonality results and connecting circuit complexity with number-theoretic properties.

Contribution

It introduces new orthogonality results between bounded depth circuit-computable functions and the Mobius function, linking circuit complexity to number theory.

Findings

Functions computed by bounded depth circuits are orthogonal to the Mobius function.

The proof combines techniques from circuit complexity and number theory.

Results relate circuit depth to the difficulty of computing number-theoretic functions.

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 Mobius function mu in the sense that E_{0 <= x <= N-1} mu(x)F(x) = o(1). The proof combines a result of Linial, Mansour and Nisan with techniques of Katai and Harman-Katai, used in their work on finding primes with specified digits.