Can you combine integration by parts and the infinite product that satisfies the exponential and Möbius function? I would like to know if is it possible to combine in a nice way the infinite product that satisifies the Möbius function and the formula for "general integration by parts".
Motivation. Since one can define for each integer $k\geq 1$ and $ \left| x \right| <1$
$$f_k(x)= \left( 1-x^k \right)^{\frac{\mu(k)}{k}} ,$$ where $\mu(n)$ is the Möbius function and we know from this MSE, Really advanced techniques of integration the shape of a "formula for general integration by parts". I think that could be possible/feasible to combine this formula with the infnite product $(16)$ of previous MathWorld's article, that is the infinite product that satisfies the Möbius function  to get a limit $\lim_{n\to\infty}$ from the formula of integration by parts, that provide us a statement.

Question. What could be a nice combination of previous statements? Can you  provide me details and hints, or hints and your final statements? I believe that could be a nice exercise. Many thanks.

I know also that, if there are no mistakes in this calculation that $$f'_k(x)= \mu(k)\left( 1-x^k \right)^{\frac{\mu(k)}{k}}\frac{x^{k-1}}{x^k-1}. $$
Final remark. I don't know if also is possible to combine with Lambert series for the Möbius function, the Prime Number Theorem... I don't know and I am saying this as motivation, thus only is required to combine the infinite product with integration by parts.
Rerences:
Möbius, Über eine besondere Art von Umkehrung der Reihen,
Journal für die reine und angewandte Mathematik (1832), Vol. 9, page 120.

Please stop with those kind of questions (which doesn't mean anything, except that you didn't understand some formula and that you want to apply it anyway)
$$-\sum_{n=1}^\infty \frac{\mu(n)}{n}\log (1-x^n) = x \quad \implies \quad \prod_{n=1}^\infty (1-x^n)^{\mu(n)/n} = e^{-x}$$ is just the definition of $\mu(n)$ together with $-\log (1-x^n) = \sum_{k=1}^\infty \frac{x^{nk}}{k}$
What to say more ? You want to play with this and integration by parts ? Ok have fun, but this isn't the way you'll learn some maths.
