Number of squarefree numbers and the Basel problem

Who discovered/proved that there are about $$\frac{x}{\zeta(2)}$$ squarefree numbers up to $x$, or (roughly) when was this first known? Today I think this is considered 'obvious', but I don't know if that was always the case.

Was this known before the Basel problem was solved (showing that the sum is $\pi^2/6$)? Perhaps it was known in the form $$x\cdot \prod_{n\ge2}\left(1+\frac{\mu(n)}{n^2}\right)$$ ?

• Preety sure that Euler was the first to prove $\zeta(2)=\pi^2/6$ which would predate the Möbius function. – Tim Raczkowski May 22 '15 at 17:18
• @TimRaczkowski: Yes, but I'm not asking about when $\zeta(2)=\pi^2/6$ was computed but rather when its application to squarefree numbers was discovered. Did Euler know about that connection when he did his work? – Charles May 22 '15 at 17:35
• @Charles: Given the fact that Euler was the first to write the $\zeta$ function as a product over the primes, I'd venture to say yes. – Lucian May 22 '15 at 18:13