TITLE: If a Dirichlet series converges Conditionally, how can I apply Euler product?
QUESTION [5 upvotes]: In 1737, Euler discovered that if $ f(n) $ is multiplicative and $ \sum f(n)/n^{s} $ converges absolutely for ${\rm Re}(s) > \sigma_a$ then we have
\begin{equation}
\sum_{n=1}^{\infty} \frac{f(n)}{n^s} ~=~ \prod_p \Bigg\{ 1+\frac{f(p)}{p^s}+\frac{f(p^2)}{p^{2s}}+ \cdots \Bigg\} 
\end{equation}
and, especially, if $f$ is completely multiplicative we have
\begin{equation}
\sum_{n=1}^{\infty} \frac{f(n)}{n^s} ~=~ \prod_p \frac{1}{1-f(p)/p^{s}}~~~~~{\rm if} ~~{\rm Re}(s)>\sigma_a.
\end{equation}
I found an example in Wikipedia (https://en.wikipedia.org/wiki/Euler_product) like
\begin{equation}
\frac{\pi}{4}~=~ \sum_{n=1}^{\infty} \frac{f(n)}{n},~~~~~~~~{\rm where}~~~~f(n)=\begin{cases}(-1)^{(n-1)/2} & {\rm if} \  n \ {\rm odd}, \\ 0 & {\rm if } \ n \ {\rm even}, \end{cases}
\end{equation}
so the theorem gives
\begin{equation}
\frac{\pi}{4}~=~ \prod_{p \not= 2} \frac{1}{1-f(p)/p} ~=~ \prod_{p\not=2} \frac{p}{p-(-1)^{(p-1)/2}}~=~\frac{3}{4}\cdot\frac{5}{4}\cdot\frac{7}{8}\cdot\frac{11}{12}\cdot \frac{13}{12}\cdots.
\end{equation}
However, this example does not converge absolutely but conditionally. 
If this example holds, how can I prove it though it converges conditionally? 
Is there any other additional condition needed or should I apply a different method?

REPLY [7 votes]: You are right to question this.  The product
$\prod_p \left(1 - \chi(p)/p\right)^{-1}$
(where $\chi = (-1/\cdot)$ is the Dirichlet character mod $4$)
does converge, and the limit is $L(1,\chi) = \pi/4$ as expected;
But this requires justification $-$ indeed it is equivalent to
the non-vanishing of the Dirichlet function $L(s,\chi)$
on the edge $s = 1+it$ of the critical strip, which is also what you need
to prove the analogue of the Prime Number Theorem for primes in arithmetic
progressions mod $4$.  (Taking logarithms, we see that 
$\prod_p \left(1 - \chi(p)/p\right)^{-1}$ converges if and only if
$\sum_p \chi(p)/p$ converges, since this sum differs from
the product's logarithm by an absolutely convergent sum $\sum_p O(1/p^2)$;
getting from $\sum_p \chi(p)/p$ to $L(s,\chi)$, and then showing that
the product $\prod_p \left(1 - \chi(p)/p\right)^{-1}$ actually converges
to $L(1,\chi)$, is a classical chapter of analytic number theory.)