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I ran across these two problems, while reading a text on number theory. The problem states: "Verify the following identities". What does "verify" mean in this context, and what strategies can I employ in trying to solve these questions?

1.) $\zeta(s)^{-1}= \sum^{\infty}_{n=1}\mu(n)/n^{s}$

2.) $\zeta(s)^{2}= \sum^{\infty}_{n=1}\nu(n)/n^{s}$

where $\zeta(s)=\sum^{\infty}_{n=1}1/n^{s}$ is the Riemann $\zeta$ function, $\mu(n)$ is the Möbius $\mu$ function and $\nu(n)$ counts the number of positive divisors of n.

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Your first question was answered here. As anon explains in his answer, Dirichlet convolution is one of the simpler ways to verify your given identities. –  J. M. Jan 26 '12 at 7:41

1 Answer 1

up vote 5 down vote accepted

"Verify" likely means perform some kind of symbolic computation that checks that the identities are in fact true. As for strategies, there is the route of direct multiplication of Dirichlet series:

$$\left(\sum_{n=1}^\infty a_nn^{-s}\right)\left(\sum_{m=1}^\infty b_mm^{-s}\right)=\sum_{n,m=1}^\infty a_nb_m(nm)^{-s}\;\stackrel{k=nm}{=}\; \sum_{k=1}^\infty \left(\sum_{nm=k}a_nb_m\right)k^{-s}.$$

Note that the coefficients above may be rewritten as $\sum_{d|k}a_db_{k/d}$. So if you have access to the fact

$$\sum_{d|k}\mu(d)\cdot1=\begin{cases}1&k=1\\0&k>1,\end{cases}$$

then you're golden. You'll also need the fact that $v(n)=\sum_{d|n}1$, naturally. Otherwise, for the first problem anyway, you can use the Euler product, viz.

$$\zeta(s)=\prod_p\left(1-\frac{1}{p^s}\right)^{-1}\implies\zeta(s)^{-1}=\prod_p\left(1-\frac{1}{p^s}\right)$$

$$=\sum_{n\text{ squarefree}}(-1)^{\omega(n)}n^{-s}=\sum_{n=1}^\infty \mu(n)n^{-s}.$$

(This is sort of hand-wavy though.)

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