A reference for $\displaystyle \sum_{n\leq X}\mu^2(n)= \frac{X}{\zeta(2)}+O(\sqrt{X}\exp(-c\sqrt{\log X}))$

Question

There is this result of counting the number of square-free numbers until $X$, which goes like this:

$$\sum_{n\leq X}\mu^2(n)= \frac{X}{\zeta(2)}+O(\sqrt{X}\exp(-c\sqrt{\log X})).$$

Could someone please point me to a reference which contains this result? I'd like to cite it and would prefer to refer the reader to a book or a paper. Thanks!

Answer 1

The specific reference mentioned by Tuvasbien is

Walfisz, Arnold. Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte, XV. VEB Deutscher Verlag der Wissenschaften, Berlin 1963.

The result quoted by Tuvasbien is Equation 27 on page 198. I don't know of an English reference for this result.