On some arithmetic properties of certain quadratic fibrations
type: Item
instance of: scholarly article
title: On some arithmetic properties of certain quadratic fibrations
zbMATH Open document ID: 0682.10018
author: Takashi Ono
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11E99
zbMATH DE Number: 4116621
zbMATH Keywords: quadratic map whose fibres are circles
reviewed by: Masakazu Muro
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227826
OpenAlex ID: W2170489288
DOI: 10.2748/TMJ/1178227826
description: scientific article; zbMATH DE number 4116621