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I have two questions regarding Non-holomorphic Eisenstein series:

$E_s(\tau)=\sum_{m,n}\frac{Im(\tau)^s}{|m+n \tau|^{2s}}$

where the sum runs over all the integers and we exclude $(0,0)$. The questions are:

1.- Does anyone know the expression for $E_s(i)$? I feel this should be known.

2.- Are the $E_s$ orthogonal with respect to some inner product? one problem seems to be that they don't decay at $\tau=i \infty$.

Thank you very much!

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Hecke and Maass already knew that (your normalization of) $E_s(i)$ is (4 times) the Dedekind zeta function of the Gaussian integers $\mathbb Z[i]$ evaluated at $s$.

You are correct that these Eisenstein series are not in $L^2$ for the $SL_2(\mathbb R)$-invariant measure $dx\,dy/y^2$ descended to the (finite-volume) quotient $SL_2(\mathbb Z)\backslash \mathfrak H$. Nevertheless, if their constant terms (0th Fourier coefficients) are truncated at any particular fixed height $y=a$, the resulting truncated Eisenstein series have inner products with each other given by the Maass-Selberg relations.

Yes, the non-truncated Eisenstein series have differing eigenvalues for the invariant Laplacian, but, not being in $L^2$, there's no sensible "orthogonality" to be proven.

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  • $\begingroup$ Thank you so much! really helpful. Do you know where can I find a reference to understand better Dedekind zeta function of the Gaussian integers? $\endgroup$
    – fernando
    Commented Oct 28, 2013 at 8:59
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    $\begingroup$ (You're welcome.) The Dedekind zeta of $\mathbb Z[i]$ is treated many places. One easily accessible place is in my notes math.umn.edu/~garrett/m/mfms/notes_c/analytic_continuation.pdf Some other notes near that link on the same page, math.umn.edu/~garrett/m/mfms discuss other related "classical" issues. $\endgroup$
    – paul garrett
    Commented Oct 28, 2013 at 11:57

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