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This question arose from a problem in Niven & Zuckerman's book "Introduction to the Theory of Numbers". In the chapter that the authors introduce the mobius function, the first exercise is the following:

Find a positive integer $n$ such that $\mu(n)+\mu(n+1)+\mu(n+2)=3$, i.e, $\mu(n)=\mu(n+1)=\mu(n+2)=1$. A brute-force aproach reveals the solution $n=33$.

My question is simply: Are there infinitely many $n$ in the previous conditions?

I really don't know how to aproach this problem. I tried various things (factorials, chinese remainder theorem, etc.) and i didn't come up with nothing. Also, for the first thousand numbers, there are the solutions $n=33,85,93,141,201,213,217,301,393,445,633,697,869,921$. One can also think in the variation of the problem with $\mu(n)=\mu(n+1)=\mu(n+2)=-1$ or simply $\mu(n)=\mu(n+1)=1$.

Thanks in advance!

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General conjectures of Chowla predict cancelation in correlations of the Mobius function: e.g. in $\sum_{n\le x}\mu(n)\mu(n+1)$ or $\sum_{n\le x} \mu(n)^2 \mu(n+1)\mu(n+2)$ etc. These conjectures would imply an asymptotic formula for the number of solutions to $\mu(n)=\mu(n+1)=\mu(n+2)$. However the conjectures are widely held to be difficult, comparable to the Hardy-Littlewood conjectures for primes.

You only ask for infinitely many values of $n$. Maybe there is a trick to doing this. For Mobius I haven't seen such a result, but for the closely related Liouville function $\lambda(n)$ (which is $-1$ if the total number of prime factors of $n$ is odd, and $1$ if it is even) such results are known. Schur first showed that $\lambda(n)=\lambda(n+1)$ happens infinitely often (and more generally for any multiplicative function taking the values $\pm 1$, with two exceptions). Hildebrand generalized this to show that all eight possible signs for $\lambda(n)$, $\lambda(n+1)$, $\lambda(n+2)$ occur infinitely often. See http://retro.seals.ch/digbib/view?rid=ensmat-001:1986:32::378&id=&id2=&id3= for Hildebrand's paper in L'Enseign. Math. Some other related work is due to Harman, Pintz and Wolke (A note on the Mobius and Liouville functions), and Buttkewitz and Elsholtz (Patterns and complexity of multiplicative functions JLMS 2011), and also work of Graham, Goldston, Pintz, and Yildirim (http://arxiv.org/abs/0803.2636 ).

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  • $\begingroup$ The link for Hildebrand's paper in L'Enseign sadly returns error 500. $\endgroup$
    – Vepir
    Commented Jun 4, 2020 at 2:58
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In a comment thread on his blog, Terry Tao writes

I don't see a way to prevent the absurd possibility that $\mu$ alternates in sign on every connected block of squarefree numbers [after some finite point].

The phrase in square brackets is my own but is implied by context; clearly, the existence of $(33,34,45)$ shows that $\mu$ does not alternate on every connected block of square free numbers.

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  • $\begingroup$ I thus understand that no non-trivial estimates are known for sums of the form $\sum_{n\le x} \mu(P(n))$, where $P$ is a polynomial of degree ${\rm deg}(P)\ge2$? $\endgroup$
    – Seva
    Commented Nov 18, 2013 at 15:25
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    $\begingroup$ That would be my guess, but I'm not a number-theorist, just someone who reads blogs by number theorists. $\endgroup$
    – David E Speyer
    Commented Nov 18, 2013 at 15:44
  • $\begingroup$ I'll comment that for the special case of the identity polynomial, this is the Mertens function. See en.wikipedia.org/wiki/Mertens_function . $\endgroup$
    – Russ Woodroofe
    Commented Nov 18, 2013 at 19:52

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