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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

11F55 Other groups and their modular and automorphic forms (several variables)
Full Text: DOI
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