Analytic formulas for the regulator of a number field

SummaryLetR=Rk andw=wk be the regulator and the number of roots of unity in the number fieldk. We determine allk for whichR/w<1/8. There are nine such fields. Sincew≥2, we have allk withR<1/4. In particular, we find the first three minima ofR over allk. There are three main ingredients to the proof:i)A new analytic formula (Theorem A below).ii)A refinement of Remak's geometric approach.iii)A refinement of Zimmert's analytic method.

[1]  F. Diaz.  Discriminant Minimal Et Petits Discriminants Des Corps De Nombres De Degre 7 Avec Cinq Places Reelles , 1988 .

[2]  J. Silverman.  An inequality relating the regulator and the discriminant of a number field , 1984 .

[3]  J. Martinet,et al.  Sur Les Minorations Géométriques Des Régulateurs , 1990 .

[4]  Rainer Zimmert,et al.  Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung , 1980 .

[5]  E. Friedman.  Hecke's integral formula , 1988 .

[6]  C. W. Clenshaw,et al.  The special functions and their approximations , 1972 .

[7]  Michael Pohst,et al.  Algorithmic algebraic number theory , 1989, Encyclopedia of mathematics and its applications.

[8]  Jacques Martinet,et al.  Journées Arithmétiques 1980: Petits discriminants des corps de nombres , 1982 .

[9]  David W. Boyd,et al.  Reciprocal polynomials having small measure. II , 1980 .

[10]  F. Diaz y Diaz,et al.  Petits discriminants des corps de nombres totalement imaginaires de degré 8 , 1987 .

[11]  H. Zassenhaus,et al.  On effective computation of fundamental units. II , 1982 .

[12]  B. Braaksma,et al.  Asymptotic expansions and analytic continuations for a class of Barnes-integrals , 1964 .

[13]  C. A. Rogers,et al.  An Introduction to the Geometry of Numbers , 1959 .

[14]  Edward Dobrowolski,et al.  On a question of Lehmer and the number of irreducible factors of a polynomial , 1979 .

[15]  S. Lang.  Algebraic Number Theory , 1971 .

[16]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[17]  W. Rheinboldt,et al.  Generalized hypergeometric functions , 1968 .

[18]  André Weil,et al.  Basic number theory , 1967 .

[19]  R. Remak.  Über Größenbeziehungen zwischen Diskriminante und Regulator eines algebraischen Zahlkörpers , 1952 .