Salem numbers of negative trace
Author:
C. J. Smyth
Journal:
Math. Comp. 69 (2000), 827838
MSC (1991):
Primary 11R06
Published electronically:
March 10, 1999
MathSciNet review:
1648407
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: We prove that, for all , there are Salem numbers of degree and trace , and that the number of such Salem numbers is . As a consequence, it follows that the number of totally positive algebraic integers of degree and trace is also .
 [BDGPS]
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Christopher
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C.J. Smyth, Cyclotomic factors of reciprocal polynomials and totally positive algebraic integers of small trace, University of Edinburgh preprint, MS96024, 1996.
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C.J. Smyth, A Euclidean algorithm for finding the intersection points of plane curves (in preparation).
 [BDGPS]
 M.J. Bertin, A. DecompsGuilloux, M. GrandetHugot, M. PathiauxDelefosse and J.P. Schreiber, Pisot and Salem numbers, Birkhäuser Verlag, Basel, 1992. MR 93k:11095
 [B]
 D.W. Boyd, Small Salem numbers, Duke Math. J. 44, (1977), 315327. MR 56:11952
 [HR]
 H. Halberstam and H.E. Richert, Sieve methods, Academic Press, London, 1974. MR 54:12689
 [MRS]
 J.F. McKee, P. Rowlinson and C.J. Smyth, Salem numbers and Pisot numbers from stars, in: Number Theory in Progress: Proceedings of the International Conference on Number Theory in Honor of Andrzej Schinzel, held in Zakopane, Poland, June 30July 9, 1997 (K. Györy, Editor), de Gruyter, Berlin, 1999, Vol. 1, 309319.
 [MSC]
 D.S. Mitrinovi\'{c}, J. Sándor and B. Crstici, Handbook of Number Theory, Kluwer, Dordrecht, 1996. MR 97f:11001
 [Robin]
 G. Robin, Estimation de la fonction de Tchebychef sur le kième nombre premier et grandes valeurs de la fonction nombre de diviseurs premiers de , Acta Arith. 42 (1983), 367389. MR 85j:11109
 [Robins]
 R.M. Robinson, Intervals containing infinitely many conjugate sets of algebraic integers, Studies in Mathematical Analysis and Related Topics, Stanford University Press, 1962, 305315. MR 26:2433
 [RS]
 J.B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math 6, (1962), 6494. MR 25:1139
 [Sa]
 R. Salem, Power series with integer coefficients, Duke Math J., 12 (1945), 153172. MR 6:206b
 [Sm1]
 C.J. Smyth, Totally positive algebraic integers of small trace, Annales de l'Institute Fourier de l'Univ. de Grenoble, 34 (1984), 128. MR 86f:11091
 [Sm2]
 C.J. Smyth, Cyclotomic factors of reciprocal polynomials and totally positive algebraic integers of small trace, University of Edinburgh preprint, MS96024, 1996.
 [Sm3]
 C.J. Smyth, A Euclidean algorithm for finding the intersection points of plane curves (in preparation).
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Additional Information
C. J. Smyth
Affiliation:
Department of Mathematics and Statistics, James Clerk Maxwell Building, King’s Buildings, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, Scotland, UK.
Email:
chris@maths.ed.ac.uk
DOI:
http://dx.doi.org/10.1090/S0025571899010996
PII:
S 00255718(99)010996
Received by editor(s):
April 28, 1998
Published electronically:
March 10, 1999
Article copyright:
© Copyright 2000 American Mathematical Society
