Number of lattice points in the hyperbolic cross
For \(s\in \mathbb{N}\) with \(s\geq 2\) let \(\Lambda\subset \mathbb{R}^s\) be a complete lattice. For a real parameter \(T>0\) the domain \(K(T):= \{\underline{x}\in \mathbb{R}^s: \overline{x}_1 \overline{x_2}\cdots \overline{x}_s\leq T\}\), where \(\overline{\xi}:= \max \{1,|\xi|\}\) for all \(\xi\in \mathbb{R}\), is called a hyperbolic cross. The authors' target is an asymptotic formula for \[ D(T;\Lambda):= \sum_{\substack{ \overline{x}_1 \overline{x}_2\cdots \overline{x}_s\leq T\\ \underline{x}\in \Lambda}} 1, \] the number of nonzero points of the lattice \(\Lambda\) in the hyperbolic cross \(K(T)\). Their result reads as follows: Theorem. Let \(T\geq 3\). Then \[ D(T;\Lambda)= \frac{2^s}{(s-1)!\det \Lambda} T\ln^{s-1} T+O \biggl( \frac{C(\Lambda)} {\det\Lambda} T \ln^{s-2}T \biggr), \] where \(C(\Lambda):= 2^s e(a_0+ 2)^s\), \(e\) Euler's constant, \(a_0\) a certain lattice constant. The implied \(O\)-constant is less or equal 1. The proof relies upon standard techniques evaluating certain \(s\)-dimensional integrals by using induction (on \(s\)).
scientific article; zbMATH DE number 1262558