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Sums of squares in \(\mathbb{Z}[\sqrt k]\). (English) Zbl 0924.11081

Author’s abstract: We study a generalization of the classical circle problem to real quadratic rings. Namely we study \[ C(N,M) = \sum_{n \leq N} \sum_{m \leq M} r(n+m \sqrt{k}) , \] where \(r(n+m \sqrt{k})\) is the number of representations of \(n+ m \sqrt{k}\) as a sum of two squares in \({\mathbb{Z}} [ \sqrt{k} ]\) (with \(k>1\) and squarefree). Using spectral theory in \(PSL_2({\mathbb{Z})}\setminus{\mathbb{H}}\), we get an asymptotic formula with error term for \(C(N,M)\), showing that some techniques on the estimation of automorphic \(L\)-functions can be applied to get upper bounds of the error term.
Reviewer: J.Hinz (Marburg)

MSC:

11P05 Waring’s problem and variants
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11E25 Sums of squares and representations by other particular quadratic forms
11N75 Applications of automorphic functions and forms to multiplicative problems
11N37 Asymptotic results on arithmetic functions
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