scholarly journals A GENERALISED KUMMER'S CONJECTURE

2010 ◽  
Vol 52 (3) ◽  
pp. 453-472 ◽  
Author(s):  
M. J. R. MYERS
Keyword(s):  
Natural Number ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Rate Of Growth ◽  
Relative Class ◽  
Almost All

AbstractKummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott–Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n.

scholarly journals The work of K. Ramachandra in algebraic number theory

2013 ◽  
Vol Volume 34-35 ◽  
Author(s):  
M. Ram Murty
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Cyclotomic Field ◽  
Algebraic Number Theory ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Relative Class ◽  
International Audience ◽  
Abelian Fields ◽  
Fundamental Units

International audience We give a brief survey of three papers of K. Ramachandra in algebraic number theory. The first paper is based on his thesis and appeared in the Annals of Mathematics and titled, ``Some Applications of Kronecker's Limit Formula.'' The second paper determines a system of fundamental units for the cyclotomic field and is titled, ``On the units of cyclotomic fields.'' This appeared in Acta Arithmetica. The third deals with relative class numbers and is titled, ``The class number of relative abelian fields.'' This appeared in Crelle's Journal.

Upper bounds on relative class numbers of cyclotomic fields

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
S. Louboutin
Keyword(s):  
Number Fields ◽  
Upper Bounds ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Mean Square ◽  
Explicit Formulas ◽  
Relative Class ◽  
P 16 ◽  
The Mean ◽  
Cyclotomic Number Fields

AbstractWe explain how one can use the explicit formulas for the mean square values of L-functions which we established elsewhere to obtain explcit upper bounds on relative class numbers of cyclotomic number fields. As an example, we show that the relative class numbers of the cyclotomic fields of conductor 4p, p ≥ 3 a prime, are less than or equal to 8√p(p/16)(p−1)/2.

scholarly journals Class numbers of cyclotomic fields

1985 ◽  
Vol 21 (3) ◽  
pp. 260-274 ◽  
Author(s):  
Gary Cornell ◽  
Lawrence C Washington
Keyword(s):  
Class Numbers ◽  
Cyclotomic Fields

scholarly journals The mean values of Dirichlet L-functions at integer points and class numbers of cyclotomic fields

1994 ◽  
Vol 134 ◽  
pp. 151-172 ◽  
Author(s):  
Masanori Katsurada ◽  
Kohji Matsumoto
Keyword(s):  
Positive Integer ◽  
Dirichlet Character ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Mean Values ◽  
Integer Points ◽  
Principal Character ◽  
Dirichlet Characters ◽  
The Mean

Let q be a positive integer, and L(s, χ) the Dirichlet L-function corresponding to a Dirichlet character χ mod q. We putwhere χ runs over all Dirichlet characters mod q except for the principal character χ0.

scholarly journals Class numbers of real cyclotomic fields of composite conductor

2014 ◽  
Vol 17 (A) ◽  
pp. 404-417 ◽  
Author(s):  
John C. Miller
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Upper Bounds ◽  
Class Numbers ◽  
Prime Ideals ◽  
Cyclotomic Fields ◽  
Class Field ◽  
Divisibility Properties ◽  
Composite Conductor ◽  
Hilbert Class Field

AbstractUntil recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.In this paper, we study in particular the cyclotomic fields of composite conductor.

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