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I am interested in the convergence of the following Euler product: $$ \prod_p \frac{1}{1-\chi(p)\cdot p^{-s}}. $$ The product is over all primes (in increasing order), with $\chi(p)=+1$ if $p \bmod 4 =...

I'm crossposting this from math.stackexchange because I think it might be inappropriately research-level for the community over there. Suppose we have an Euler product over the primes $$F(s) = \prod_{...

It is well known that $$\prod_p\,(1-p^{-1})=\frac 1 {\zeta(1)}=0$$ Given an arbitrary prime $\,q\,$ is it true that $$\prod_{q\,|\,p+1}\,(1-p^{-1})=0\;\;\;?$$ Thanks.

Is anything useful known about the function defined by \[ f(s, \alpha) = \prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1} \quad ? \] Here, $\alpha$ is real. When $\alpha = 1$, this is certainly the ...

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)}{...

This question expands on this one from MSE. In the literature about Dirichlet $L$-series, I found that their Euler products: $$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$ ...