Show the function for which the Dirichlet generating series is $\zeta(2s)$ using only $\tau,\varphi,\sigma\text{ and }\mu$ or some explicit formula.

Question

I'm trying to find the function with Dirichlet generating series $\zeta(2s)$, I know that this relates somehow to the Liouville function but I am trying to express it in terms of only the standard arithmetic functions $\varphi,\tau,\sigma,\mu$ or some explicit formula. What I have tried so far is I know,

$$F_f(s)=\sum_{j=1}^\infty\frac{f(j)}{j^s}$$ $$F_f(s-1)=\sum_{j=1}^\infty\frac{jf(j)}{j^s}$$

And I have tried to find the g such that

$$F_g(s)=\sum_{j=1}^\infty\frac{g(j)}{j^{2s}}$$

But everything I have tried from this point onwards has been unsuccessful. Any help would be great.

Answer 1

In this sort of question, it is often a good idea to consider the effect on the Euler product: $$ \zeta(2s) = \prod_p \left( 1 - \frac{1}{p^{2s}} \right)^{-1}. $$ This particular question happens to be relatively straightforward, as this is obviously multiplicative and this expression indicates that the coefficients are defined on prime powers by the function $$ a(p^k) = \begin{cases} 1 & 2 \mid k \\ 0 & \text{else}. \end{cases} $$ This is, of course, another name for the "is a square" function.

Visually inspecting, we can quickly convince ourselves of this as well: $$ \zeta(2s) = 1 + \frac{1}{4^s} + \frac{1}{9^s} + \frac{1}{16^s} + \frac{1}{25^s} + \cdots$$