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I've been reading about Shintani zeta functions and in particular with respect to finding the density of cubic discriminants as in the theorem of Davenport-Heilbronn. In Shintani's paper "On zeta-functions associated with the vector space of quadratic forms" [Tokyo Univ. J. Fac. Sci Sect. 1A Math 1975], in the proof of Theorem 4, Shintani writes

Hence, we have (by earlier results in this paper and in a previous paper): $$\sum_{nk^4 \leq x} h_r(n) = 2^{-1}\zeta(2)\zeta(4)x + O(x^{2/3 + \epsilon}) \qquad (x \to +\infty, \forall \epsilon > 0)$$ An application of the Mobius inversion formula now yields that
$$\sum_{n \leq x} h_r(n) = 2^{-1}\zeta(2)x + O(x^{2/3 + \epsilon})$$

When I think of Mobius Inversion, I think of two things: \begin{align*} g(n) &= \sum_{d \mid n} f(n/d) \iff f(n) = \sum_{d \mid n} \mu(n)g(n/d) \quad \text{or}\\ g(x) &= \sum_{n \leq x}f(n/x) \iff f(x) = \sum_{n \leq x} \mu(n)g(n/x) \end{align*}

But I don't see how I can use these here. Unfortunately, I also know that there are many things that might be called Mobius Inversion. This is one of those steps that taunts me. Qualitatively, we remove the fourth-power condition and end up losing a factor of $\zeta(4)$, and that feels very reasonable.

Further, in playing with it for a while, I re-stumbled upon the fact that $\displaystyle \dfrac{1}{\zeta(s)} = \sum_n \dfrac{\mu(n)}{n^s}$ (easy, but which I did not originally remember) and thus that $\displaystyle \dfrac{1}{\zeta(4)} = \sum \dfrac{\mu(n)}{n^4}$.

To make my question explicit - how can we arrive at the second equation from the first?

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# An application of Mobius Inversion in a paper of Shintani

I've been reading about Shintani zeta functions and in particular with respect to finding the density of cubic discriminants as in the theorem of Davenport-Heilbronn. In Shintani's paper "On zeta-functions associated with the vector space of quadratic forms" [Tokyo Univ. J. Fac. Sci Sect. 1A Math 1975], in the proof of Theorem 4, Shintani writes

Hence, we have (by earlier results in this paper and in a previous paper): $$\sum_{nk^4 \leq x} h_r(n) = 2^{-1}\zeta(2)\zeta(4)x + O(x^{2/3 + \epsilon}) \qquad (x \to +\infty, \forall \epsilon > 0)$$ An application of the Mobius inversion formula now yields that
$$\sum_{n \leq x} h_r(n) = 2^{-1}\zeta(2)x + O(x^{2/3 + \epsilon})$$

When I think of Mobius Inversion, I think of two things: \begin{align*} g(n) &= \sum_{d \mid n} f(n/d) \iff f(n) = \sum_{d \mid n} \mu(n)g(n/d) \quad \text{or}\\ g(x) &= \sum_{n \leq x}f(n/x) \iff f(x) = \sum_{n \leq x} \mu(n)g(n/x) \end{align*}

But I don't see how I can use these here. Unfortunately, I also know that there are many things that might be called Mobius Inversion. This is one of those steps that taunts me. Qualitatively, we remove the fourth-power condition and end up losing a factor of $\zeta(4)$, and that feels very reasonable.

Further, in playing with it for a while, I re-stumbled upon the fact that $\displaystyle \dfrac{1}{\zeta(s)} = \sum_n \dfrac{\mu(n)}{n^s}$ (easy, but which I did not originally remember) and thus that $\displaystyle \dfrac{1}{\zeta(4)} = \sum \dfrac{\mu(n)}{n^4}$.