Best known bounds for a product over primes in an interval As the title ask, what is the best lower and upper bounds for the product below :
$$ \prod \limits_{x < p \leq y} \frac{p+1}{p}$$
such that $p$ denote the prime numbers in which fullfil the conditions under the product ?
 A: Your product is the following $$\prod_{y<p\leq x} \left(1+\frac{1}{p}\right).$$
You can use the fact that $$\prod_{y<p\leq x} \left(1+\frac{1}{p}\right)\leq \prod_{y<p\leq x} \left(1-\frac{1}{p}\right)^{-1} \leq \frac{\pi^2}{6} \prod_{y<p\leq x} \left(1+\frac{1}{p}\right).$$ Note that $\frac{\pi^2}{6}=\frac{1}{\zeta(2)},$ where $\zeta$ is the Riemann zeta function. According to the best current explicit results of Pierre Dusart: For $x > 1$,
$$\prod_{p\leq x} \left(1-\frac{1}{p}\right)< \frac{\exp(-\gamma)}{\ln{x}} \left(1+\frac{0.2}{\ln^3{x}}\right)$$
and for $x \geq 2278382,$ 
$$\prod_{p\leq x} \left(1-\frac{1}{p}\right)> \frac{\exp(-\gamma)}{\ln{x}} \left(1-\frac{0.2}{\ln^3{x}}\right)$$
where $\gamma$ is the Euler's constant $(\gamma \sim 0.5772157)$  and you are done! 
A: The best source for explicit, unconditional bounds on such products that I'm aware of is in the work of Pierre Dusart.  See his paper Explicit estimates of some functions over primes.
In Section 5.4, Theorem 5.9, he gives upper and lower bounds for
$$
Q(x):=\prod_{p\leq x}\frac{p}{p-1}.
$$
Using these, we can get bounds on
$$
Q(y)/Q(x)=\prod_{x< p\leq y}\frac{p}{p-1}.
$$
Notice that the fraction you want is
$$
\frac{p+1}{p}=\frac{p}{p-1}\cdot \left(1-\frac{1}{p^2}\right)
$$
so you have reduced to bounding
$$
\prod_{x<p\leq y}\left(1-\frac{1}{p^2}\right)
$$
which can be done by standard techniques.
(Note that Dusart's bounds are for $x\geq 2278382$, but for small values of $x$ and $y$ you can do an explicit computation.)  Alternatively, you could work directly with the fraction you want (instead of $p/(p-1)$) and use the same techniques as Dusart.
A: Though Dusart bounds are good ones for $\,\prod\limits_{p \le x}{\left(1 -1/p\right)}$, I think a better idea to the prime product asked by Ahmad is to explore the Dedekind psi function, used in the theory of modular functions, defined as $\,\psi(n)/n = \prod\limits_{p|n}\left( 1 +\frac{1}{p} \right)$.
