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**Question:** can

$$

\zeta_1(s)

= \prod_{p \equiv 3 \pmod{4}} \frac{1}{1 – p^{-s}}

$$

be evaluated or written in terms of standard functions?

**Details:**

We can write the Riemann zeta function as

\begin{align*}

\zeta(s)

&= \prod_{p \text{ prime}} \frac{1}{1 – p^{-s}} \\

&= \frac{1}{1-2^{-s}}

\;\;

\underbrace{\prod_{p \equiv 1 \pmod{4}} \frac{1}{1 – p^{-s}}}_{\zeta_0(s)}

\quad

\underbrace{\prod_{p \equiv 3 \pmod{4}} \frac{1}{1 – p^{-s}}}_{\zeta_1(s)},

\end{align*}

so I’m asking for an evaluation of $\zeta_0(s)$ and $\zeta_1(s)$.

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I encountered this while finding an expression for the probability that a “random” integer can be written as a sum of two squares, in this answer.

Since an integer is the sum of two squares iff it has an even number of each prime factor $\equiv 3 \pmod{4}$, the answer probability can be naturally written (after some work) as

$$

\lim_{x \to 1} \frac{\zeta_1(2x)}{\zeta_1(x)}.

$$

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