I know that the Euler product of a summation of multiplicative function is given by  $$\sum_nf(n)=\prod_p (1+f(p)+f(p^2)+....),$$ and if we have the Möbius function then it will be  $$\sum_n\mu (n)f(n)=\prod_p (1+f(p)).$$

I would like to know if there is a general formula for Euler product of a double summation? I saw in one of the papers he applied the Euler product like this  $$ \sum_{a,r}\mu (a)f(a)^2\mu (r)\frac{f(r)}{\sqrt{r}}=\prod_p (1+f(p)^2+\frac{f(p)}{\sqrt{p}})$$

I know that the Möbius function vanish all other terms, but I would like to know the general one without the Mobius.

  Thanks

I know that the Euler product of a summation of multiplicative function is given by 

$$\sum_nf(n)=\prod_p (1+f(p)+f(p^2)+....),$$

and if we have the Möbius function then it will be 

$$\sum_n\mu (n)f(n)=\prod_p (1+f(p)).$$

I would like to know if there is a general formula for Euler product of a double summation? I saw in one of the papers he applied the Euler product like this 

$$ \sum_{a,r}\mu (a)f(a)^2\mu (r)\frac{f(r)}{\sqrt{r}}=\prod_p (1+f(p)^2+\frac{f(p)}{\sqrt{p}})$$

I know that the Möbius function vanish all other terms, but I would like to know the general one without the Mobius. Thanks.