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I am looking for upper and lower bounds for the following expression:

$$\prod_{\substack{p\le n \\ p \equiv 1\ mod\ 4}} \frac{p-1}{p}$$

Apart from the trivial one:

$$\prod_{\substack{p\le n \\ p \equiv 1\ mod\ 4}} \frac{p-1}{p} \ge \prod_{p\le n} \frac{p-1}{p}$$

Any advice or bibliographic reference will be highly appreciated.

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    $\begingroup$ Have you seen mathoverflow.net/questions/374824/… ? $\endgroup$
    – KConrad
    Commented Mar 29, 2023 at 5:52
  • $\begingroup$ For any $r,m\in\mathbb{N}$ with $\gcd(r,m)=1$ we have $\prod_{\substack{p\le x \\ p \equiv r\ mod\ m}} \frac{p-1}{p}\sim \frac{C}{\log(x)^{1/\phi(m)}}$ for some constant $C$. $\endgroup$
    – Ethan Splaver
    Commented Jun 6, 2023 at 7:58

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