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I'm a string theorist and I have come across the following expression in a computation I'm doing (involving a sum over inequivalent Lens spaces): $$\widehat{\zeta}(s)=\prod_{\mathrm{primes}\ p\equiv 3\ (\mathrm{mod}\ 4)}(1-p^{-s})^{-1}.$$ I expected this to be a standard sort of construction, and it's clearly closely related to certain standard zeta-functions and Dirichlet $L$-functions (for instance it seems that $\widehat{\zeta}(s)^2/\widehat{\zeta}(2s)=L(\chi_0,s)/L(\chi_1,s)$, where $\chi_0$ and $\chi_1$ are the trivial and nontrivial Dirichlet characters modulo $4$), but with my poor knowledge of this very wide field, I wasn't quite able to find this particular combination. I would like to extend this analytically in $s$ and am particularly interested in any poles along the real axis and their residues.

Any help or references would be much appreciated, and apologies if there was an easily found answer that I missed.

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@Daniel: Exercises 21 and 22 in Chapter 6.2 of Montgomery & Vaughan's "Multiplicative Number Theory" might be useful. They show how to analytically manipulate Euler products similar to $\hat{\zeta}(s)$. – Micah Milinovich Jul 14 '11 at 14:47
Thanks to both GH and Micah - your answers and the reference will be quite helpful. – Daniel Robbins Jul 14 '11 at 14:54
up vote 6 down vote accepted

Your functional equation shows that $\widehat{\zeta}(s)^2$ has a meromorphic continuation to $\Re(s)>1/2$ with a simple pole at $s=1$. This also shows that $\widehat{\zeta}(s)$ does not have a meromorphic continuation to a neighborhood of $s=1$, instead it lives on a double cover of that neighborhood branched at $s=1$.

If you want to extend further to the left, you need even higher powers of $\widehat{\zeta}(s)$: precisely the $2^k$-th power for $\Re(s)>2^{-k}$. The poles on the positive real axis will be at the points $s=2^{-k}$, for the relevant powers of $\widehat{\zeta}(s)$. This also tells me that there is no reasonable continuation to the left of the imaginary axis.

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