A remark on symplectic matrices (Q1913333)

Let \(\Gamma_g\) be the Siegel modular group of degree \(g\) with the standard subgroup \(\Gamma_{g, \infty}= \{{{*\;*} \choose {0\;*}}\in \Gamma_g\}\). \(\Gamma_g\) acts on the Siegel half-space \({\mathcal H}_g\) in the usual way. If \(h(Z):= \text{det} (\text{Im Z})\), \(Z\in {\mathcal H}_g\), denotes the height, the author investigates the arithmetic function \[ \alpha_Z (n):= \# \Bigl\{ \Gamma_{g, \infty} M;\;M\in \Gamma_g,\;{\textstyle {1\over {n+1}}} \leq h(M \langle Z\rangle)< {\textstyle {1\over n}} \Bigr\}, \] which plays a decisive role in the proof of convergence of the Siegel-Eisenstein series. The author's main result is \[ \alpha_Z (n)= O(n^{(g+ 1)/2- \delta_g+ \varepsilon}), \quad \varepsilon >0, \quad \delta_g= 1\Bigl/ \Bigl( 1+2 \bigl[ {\textstyle {g\over 2}} \bigr]+ {\textstyle {1\over {g+1}}} \Bigr). \] The proof uses analytic properties of Eisenstein series and a version of Landau's theorem.