scholarly journals Heuristics for class numbers of prime-power real cyclotomic fields

Author(s):  
Joe Buhler ◽  
Carl Pomerance ◽  
Leanne Robertson
2018 ◽  
Vol 238 ◽  
pp. 206-214 ◽  
Author(s):  
DAVID S. DUMMIT

We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal.


2010 ◽  
Vol 52 (3) ◽  
pp. 453-472 ◽  
Author(s):  
M. J. R. MYERS

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.


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

1994 ◽  
Vol 134 ◽  
pp. 151-172 ◽  
Author(s):  
Masanori Katsurada ◽  
Kohji Matsumoto

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.


2014 ◽  
Vol 17 (A) ◽  
pp. 404-417 ◽  
Author(s):  
John C. Miller

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.


2013 ◽  
Vol Volume 34-35 ◽  
Author(s):  
M. Ram Murty

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.


2006 ◽  
Vol 120 (2) ◽  
pp. 372-384 ◽  
Author(s):  
István Gaál ◽  
Leanne Robertson

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