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?


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 $\zeta_1(s)$.

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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    $\begingroup$ What do you mean by "standard function"? For example, for me $\zeta$ is a standard function, is it for you? $\endgroup$ – Wojowu Aug 30 '15 at 21:23
  • $\begingroup$ @Wojowu I mean something very broad. Of course $\zeta$ function is standard. Any other function that has been defined in the literature or in a paper is standard. $\endgroup$ – 6005 Aug 30 '15 at 21:29
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    $\begingroup$ See this MathOverflow question here: mathoverflow.net/questions/28000/… $\endgroup$ – Peter Humphries Aug 31 '15 at 19:57

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