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 $$ x\cdot \prod_{n\ge2}\left(1+\frac{\mu(n)}{n^2}\right) $$ ?

  • $\begingroup$ Preety sure that Euler was the first to prove $\zeta(2)=\pi^2/6$ which would predate the Möbius function. $\endgroup$ – Tim Raczkowski May 22 '15 at 17:18
  • $\begingroup$ @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? $\endgroup$ – Charles May 22 '15 at 17:35
  • 1
    $\begingroup$ @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. $\endgroup$ – Lucian May 22 '15 at 18:13

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