In order to know more about product over primes ,I would like to know how do I show that :$$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$$ without using properties of Riemann zeta function ?


**Note01** : it is well known that $$\prod\frac{p^2+1}{p^2-1}=\frac{{\zeta}^2(2)}{\zeta(4)}=\frac{5}{2}$$ but is there other method to show that ?


**Note 02** :I wish using divisor function properties 


Thank you for any help



How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function?
This is a well-known problem, attributed to Sam Wagstaff in Richard Guy's *Unsolved Problems in Number Theory*. Section B48 "Products taken over primes" includes a paragraph



> 
> Wagstaff asked for an elementary proof (e.g., without using properties of the Riemann zeta-function that $$\prod\_p \frac{p^2+1}{p^2-1} = \frac52 $$
>  where the product is taken over all primes. It seems very unlikely that there is a proof which doesn't involve analytical methods. At first glance it might appear that the fractions might cancel, but none of the numerators
>  are divisible by 3 [. . .]
> 
> 
> 


with a reference to



> 
> David Borwein & Jonathan M. Borwein, On an intriguing integral and some series related to $\zeta(4)$, *Proc. Amer. Math. Soc.*, **123** (1995) 1191-1198; *MR* **95e**:11137.
> 
> 
> 


