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On a certain function analogous to $$\log \vert\eta(z)\vert$$. (English) Zbl 0213.05701
Let $$k$$ be a number field with class number one and $$R$$ its ring of integers. The group $$\mathrm{SL}(2,\mathbb R)$$ acts properly discontinuously on a product $$H= H_1^{r_1}\times H_2^{r_2}$$ of complex upper half-plane $$H_1$$ and quaternionic upper half-space $$H_2$$. The author studies a non-holomorphic (in $$z\in H)$$ Eisenstein series $$E_k(z,s)$$, defined first for $$\operatorname{Re}(s) > 1$$, and its Laurent expansion at $$s=1$$. The constant term in this expansion (in powers of $$s-1)$$ contains a real valued function $$h_k(z)$$ which corresponds to $$\log\vert\eta(z)\vert^4$$ when $$k = \mathbb Q$$. In this particular case
$E_{\mathbb Q}(z,s) = \frac12 \sum{}' \frac{\operatorname{Im}(z)^s}{\vert mz + n\vert^{2s}} = \zeta(2s) \sum_{\gamma\in \Gamma_\infty\backslash \Gamma} \operatorname{Im} \gamma(z)^s$
where $$\Gamma = \mathrm{SL}(2,\mathbb Z) \supset \Gamma_\infty$$ subgroup of matrices of the form $$\begin{pmatrix} a & b\\ c & d\end{pmatrix}$$. The limit formula of Kronecker gives indeed in this case
$E_{\mathbb Q}(z,s) = \frac{\pi/2}{s-1} + \frac{\pi}{2} (2C - \log 4 - \log \operatorname{Im}(z) - \log\vert\eta(z)\vert^4 + O(s-1).$
It is shown that $$h_k(z)$$ has similar properties to $$\log\vert\eta(z)\vert^4$$ in general. In particular, it is a real valued harmonic function on $$H$$, behaves as the logarithm of the absolute value of a modular form for $$\mathrm{SL}(2,\mathbb R)$$ of weight one on $$H$$, and is the Mellin transform of $$\zeta_k(s)\zeta_k(s+1)$$, where $$\zeta_k$$ denotes the Dedekind zeta function of $$k$$. For this generalized Mellin transformation, one should compare with
Reviewer: Alain Robert

##### MSC:
 11F55 Other groups and their modular and automorphic forms (several variables)
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##### References:
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