On the distribution of the eigenvalues of the hyperbolic Laplacian for \(\operatorname {PSL} (2,\mathbb{Z})\)
Let \(0= \lambda_0< \lambda_1< \lambda_2<\cdots\) be the different eigenvalues of the hyperbolic Laplacian acting on \(\text{PSL}_2 (\mathbb{Z})\setminus \mathbb{H}\) where \(\mathbb{H}\) denotes the upper half-plane, and let \(\lambda_j= \frac 14+ t_j^2\) with \(t_j> 0\) \((j\geq 1)\). Employing the Selberg trace formula the author wants to show that the numbers \(t_j\log t_j\) \((j\geq 1)\) are rather regularly distributed \(\operatorname {mod} 2\pi\) (in some sense which is too long to be given here). Reviewer's remark: This paper contains a large number of misprints. More seriously the function \(h(t)= \frac{4} {\sqrt{2\pi}} \operatorname {Im} K_{2it} (2\pi\ell (\delta-i))\) to which Selberg's trace formula is applied is obviously not holomorphic and the pair of functions \(g,h\) is not a pair of Selberg transforms. The error is visible in the proof of Lemma 2.2, which is valid for real values of \(t\) only. Hence it is unclear to the reviewer whether the alleged theorem actually holds true. Part II of the paper under review appeared in J. Number Theory 86, 284-301 (2001; Zbl 1035.11022).
scientific article; zbMATH DE number 1488556