In this book, the author studies the Dirichlet series whose coefficients are the number of orders of a quartic field with given indices. Nakagawa gives an explicit expression of the Dirichlet series. Using this expression, its analytic properties are deduced. He also presents an asymptotic formula for the number of orders in a quartic field with index less than a given positive number.

Readership

Graduate students and research mathematicians interested in number theory, specifically cubic and quartic extensions.

Table of Contents

• Introduction
• Preliminaries
• Type \(1111\)
• Types \(112\) and \(111^2\)
• Types \(22\), \(21^2\) and \(1^21^2\)
• Types \(13\) and \(11^3\)
• Types \(4\) and \(1^4\)
• Type \(2^2\)
• Proof of Theorem 1
• Bibliography