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Salem numbers of negative trace

Author: C. J. Smyth
Journal: Math. Comp. 69 (2000), 827-838
MSC (1991): Primary 11R06
Published electronically: March 10, 1999
MathSciNet review: 1648407
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Abstract: We prove that, for all $d\geq 4$, there are Salem numbers of degree $2d$ and trace $-1$, and that the number of such Salem numbers is $\gg d/\left( \log \log d\right) ^{2}$. As a consequence, it follows that the number of totally positive algebraic integers of degree $d$ and trace $2d-1$ is also $\gg d/\left( \log \log d\right) ^{2}$.

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  • [BDGPS] M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and J.-P. Schreiber, Pisot and Salem numbers, Birkhäuser Verlag, Basel, 1992. With a preface by David W. Boyd. MR 1187044
  • [B] David W. Boyd, Small Salem numbers, Duke Math. J. 44 (1977), no. 2, 315–328. MR 0453692
  • [HR] H. Halberstam and H.-E. Richert, Sieve methods, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR 0424730
  • [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 30-July 9, 1997 (K. Györy, Editor), de Gruyter, Berlin, 1999, Vol. 1, 309-319.
  • [MSC] D. S. Mitrinović, J. Sándor, and B. Crstici, Handbook of number theory, Mathematics and its Applications, vol. 351, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1374329
  • [Robin] Guy Robin, Estimation de la fonction de Tchebychef 𝜃 sur le 𝑘-ième nombre premier et grandes valeurs de la fonction 𝜔(𝑛) nombre de diviseurs premiers de 𝑛, Acta Arith. 42 (1983), no. 4, 367–389 (French). MR 736719
  • [Robins] Raphael M. Robinson, Intervals containing infinitely many sets of conjugate algebraic integers, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 305–315. MR 0144892
  • [RS] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689
  • [Sa] R. Salem, Power series with integer coefficients, Duke Math J., 12 (1945), 153-172. MR 6:206b
  • [Sm1] Christopher Smyth, Totally positive algebraic integers of small trace, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 3, 1–28 (English, with French summary). MR 762691
  • [Sm2] C.J. Smyth, Cyclotomic factors of reciprocal polynomials and totally positive algebraic integers of small trace, University of Edinburgh preprint, MS96-024, 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.

Received by editor(s): April 28, 1998
Published electronically: March 10, 1999
Article copyright: © Copyright 2000 American Mathematical Society