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It is well known that $$\sum_{n\leq x}\mid \mu(n) \mid \sim \frac{6}{\pi^2}x\left(1+o(1)\right) \text{, } x \to \infty$$ From here it follows that every sufficient large integer may be expressed as the sum of two square-free integers. However, the methods to prove this, give no hint as to how to count these combinations. Moreover, I have not seen this problem stated anywhere!

The analogous story regarding sums of squares is more rich. For example, Jacobis four square theorem $$r_4(n)=8\sum_{k\mid n,4\nmid m}k$$ gives an explicit formula for the number of ways, $r_4(n)$, to express a given integer as the sum of four squares.

Is there a simple way to approach the problem for square free integers? Even obtaining an upper bound for $sf_2(n)$ is proving difficult.

I look forward to hearing your ideas!

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  • $\begingroup$ Re: "... follows that every sufficient large integer may be expressed as the sum of two square-free integers", note On the Sum of a Prime and a Square-free Number links to the Oct. 2014 Ramanujan article where the abstract says "We prove that every integer greater than two may be written as the sum of a prime and a square-free number." As primes are square-free integers, this is even stronger than every integer greater than $2$ may be written as a sum of $2$ square-free integers. $\endgroup$ – John Omielan Feb 7 at 6:09
  • $\begingroup$ The same author, Adrian Dudek, has another paper, from Jan. 26, 2016 at arxiv (but it may now also be published elsewhere), On the Sum of the Square of a Prime and a Square-Free Number, which provides a strong statement, although it's limited to $\ge 10$ and $n \not{\equiv} \; 1 \pmod 4$. You may wish to contact him as he likely knows a lot about sums of $2$ square-free integers. $\endgroup$ – John Omielan Feb 7 at 6:15
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Your $sf_2(n)$ is OEIS A071068, Number of ways to write n as a sum of two unordered squarefree numbers. The comments note that (with notation marked up slightly and $a(n)$ replaced with $sf_2(n)$ for clarity)

The natural density of the squarefree numbers is $\frac{6}{\pi^2}$, so $An < sf_2(n) < Bn$ for all large enough $n$ with $A < \frac{6}{\pi^2} - \frac12$ and $B > \frac{3}{\pi^2}$. The Schnirelmann density of the squarefree numbers is $\frac{53}{88} > \frac12$, and so $sf_2(n) > 0$ for all $n > 1$ (in fact, $sf_2(n+1) \ge \frac{9n}{88}$). It follows from Theoreme 3 bis. in Cohen, Dress, & El Marraki along with finite checking up to 16089908 that $0.10792n < sf_2(n) < 0.303967n$ for $n > 36$. (The lower bound holds for $n > 1$.) - Charles R Greathouse IV, Feb 02 2016

where the referenced paper is Henri Cohen, Francois Dress, and Mahomed El Marraki, Explicit estimates for summatory functions linked to the Möbius μ-function, Funct. Approx. Comment. Math. 37:1 (2007), pp. 51-63.

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  • $\begingroup$ I believe your expression of $sf_2(n)$ describes the number of ways to express a given integer as two non-squares. Are you saying that these are the same as square free numbers? $\endgroup$ – user385459 Feb 21 at 16:39
  • $\begingroup$ Answer completely rewritten. $\endgroup$ – Peter Taylor Feb 21 at 17:21

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