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Does anyone know an expression (in terms of simpler functions) for the following Epstein Zeta function:

$\sum \frac{1}{(m^2+m n+n^2)^s}$

I know an expression (in terms of the Dirichlet Beta function) exist for $\sum \frac{1}{(m^2+n^2)^s}$. I did look hard into the literature (being not an expert) but couldn't find anything.

Thank you very much!

fernando

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I had thought the question meant to refer to Dedekind zeta, not Beta, since that Epstein zeta is ($4\times$) the Dedekind zeta function of the Gaussian integers. The case of the question gives ($6\times$) the Dedekind zeta function of the "Eisenstein integers", namely, $\mathbb Z[\rho]$, where $\rho$ is a cube root of unity. To see this, one uses the fact that that ring is Euclidean, and that $N(m+n\rho)=m^2+mn+n^2$.

Edit/addition: Ah, sorry, didn't realize the reference to Dedekind "beta"... Yes, by a special case of quadratic reciprocity, the Dedekind zeta function of the Gaussian integers factors as "ordinary" zeta times (what many might call) the Dirichlet $L$-function $L(s,\chi)=\sum_n \chi(n)/n^s$ attached to the character $\chi(n)$ which is $0$ for $n=2\mod 4$, is $+1$ for $n=1\mod 4$, and is $-1$ for $n=3\mod 4$. This gives the Dedekind "beta function".

As @HenryCohn notes, in the case of $m^2+mn+n^2$, because of quadratic reciprocity for $-3$, the Epstein zeta factors again as $\zeta(s)\cdot L(s,\chi)$, now with $\chi(n)=+1$ for $n=1\mod 3$, $\chi(n)=-1$ for $n=2\mod 3$, and $\chi(n)=0$ for $n=0\mod 3$.

Those factorizations due to quadratic reciprocity were known to Dirichlet for quadratic extensions of the rationals.

Further-edit: I had forgotten: although I think it's not the most important or revealing relation to other things, yes, such Dirichlet $L$-functions are indeed expressible in terms of the Hurwitz zeta: for example, for the $m^2+mn+n^2$ case, after the factor of $\zeta(s)$ is removed, the remaining $L(s,\chi)=\sum_n\chi(n)/n^s$ is a linear combination of $\sum_n 1/(n+{1\over 3})^s$ and $\sum_n 1/(n+{2\over 3})^s$, and similarly in general.

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  • $\begingroup$ I suspect the question does mean Dirichlet beta function, which seems to be a name for the L function of the nontrivial character modulo 4 (see en.wikipedia.org/wiki/Dirichlet_beta_function). But of course there's a corresponding factorization of the Dedekind zeta function in either case, if it matters. $\endgroup$
    – Henry Cohn
    Nov 9, 2013 at 4:51
  • $\begingroup$ Thank you so much to you both. Indeed, I meant Dirichlet beta function. Sorry about the question being a bit elementary, not being a Mathematician myself sometimes is hard to judge. Do you know whether the series above (or equivalently the Dedekind zeta function of the "Eisenstein integers") can be written in terms of Hurwitz zeta functions? this is certainly the case for the quadratic form $m^2+n^2$. $\endgroup$
    – fernando
    Nov 9, 2013 at 8:49

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