On some arithmetic properties of certain quadratic fibrations
For a natural number n, consider the following two cases: \[ A(n)=\{(x,y,t,u)\in {\mathbb{Z}}^ 4;\quad \gcd (x,y,t,u)=1,\quad x^ 2+y^ 2=tu,\quad t+u=n\}, \] \[ B(n)=\{(x,y,t,u)\in {\mathbb{Z}}^ 4;\quad \gcd (x,y,t,u)=1,\quad x^ 2+y^ 2=tu=n,\quad u\geq 1\}. \] Denote by a(n), b(n) the cardinality of A(n), B(n), respectively. A proof of the following formulas is given in this paper: (1) \(\theta^ 4_ 3(\tau)=1+\sum^{\infty}_{n=1}a(n)(\theta^ 2_ 3(n\tau)-1)\) where \(\theta_ 3(\tau)=\sum_{k\in {\mathbb{Z}}}\exp (\pi i\tau k^ 2);\) (2) \(4\zeta_{{\mathbb{Q}}(i)}(s)^ 2/\zeta_{{\mathbb{Q}}(i)}(2s)=\sum^{\infty}_{n=1}b(n)n^{-s}\) where \(\zeta_{{\mathbb{Q}}(i)}(s)=(1/4)\sum_{(a,b)\in {\mathbb{Z}}^ 2}(a^ 2+b^ 2)^{-s}\) ((a,b)\(\neq (0,0)).\) These formulas are special cases of more general formulas and are proved by looking at a quadratic map f whose fibres are circles.
scientific article; zbMATH DE number 4116621