The mean values of totally real algebraic integers

Author:
C. J. Smyth

Journal:
Math. Comp. **42** (1984), 663-681

MSC:
Primary 11R80; Secondary 11R04, 11S05

DOI:
https://doi.org/10.1090/S0025-5718-1984-0736460-5

MathSciNet review:
736460

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the *p*th root of the mean absolute values of the *p*th powers of a totally real algebraic integer . For each fixed we study the set of such . We show that its structure is as follows: on the nonnegative real line it consists of some isolated points, followed by a small interval in which its structure is as yet undetermined. Beyond this small interval, it is everywhere dense.

**[1]**D. W. Boyd, "Small Salem numbers,"*Duke Math. J.*, v. 44, 1977, pp. 315-328. MR**0453692 (56:11952)****[2]**V. Ennola, "Conjugate algebraic integers in an interval,"*Proc. Amer. Math. Soc.*, v. 53, 1975, pp. 259-261. MR**0382219 (52:3104)****[3]**H. Gould,*Combinatorial Identities*, Morgantown Printing Co., 1972. MR**0354401 (50:6879)****[4]**I. S. Gradshteyn & I. M. Ryzhik,*Table of Integrals, Series, and Products*, Academic Press, New York, 1965.**[5]**D. H. Greene & D. E. Knuth,*Mathematics for the Analysis of Algorithms*, Birkhäuser, Boston, 1981.**[6]**G. H. Hardy, J. E. Littlewood & G. Polya,*Inequalities*, 2nd ed., Cambridge Univ. Press, 1952. MR**0046395 (13:727e)****[7]**J. Hunter, "A generalisation of the inequality of the arithmetic-geometric means,"*Glasgow Math. J.*, v. 1, 1958, pp. 149-158. MR**0075984 (17:828g)****[8]**L. Lewin,*Polylogarithms and Associated Functions*, North-Holland, Amsterdam, 1981. MR**618278 (83b:33019)****[9]**M. J. McAuley,*Topics in J-Fields and a Diameter Problem*, M. Sc. thesis, University of Adelaide, 1981.**[10]**D. S. Mitrinović,*Analytic Inequalities*, Springer, New York, 1970. MR**0274686 (43:448)****[11]**R. M. Robinson, "Intervals containing infinitely many sets of conjugate algebraic integers,"*Studies in Mathematical Analysis and Related Topics*:*Essays in Honor of George Polya*, Stanford, 1962, pp. 305-315. MR**0144892 (26:2433)****[12]**R. M. Robinson, "Algebraic equations with span less than 4,"*Math. Comp.*, v. 10, 1964, pp. 549-559. MR**0169374 (29:6624)****[13]**C. L. Siegel, "The trace of totally positive and real algebraic integers,"*Ann. of Math.*, v. 46, 1945, pp. 302-312. MR**0012092 (6:257a)****[14]**C. J. Smyth, "On the measure of totally real algebraic integers,"*J. Austral. Math. Soc. Ser. A*, v. 30, 1980, pp. 137-149. MR**607924 (82j:12002a)****[15]**C. J. Smyth, "On the measure of totally real algebraic integers. II,"*Math. Comp.*, v. 37, 1981, pp. 205-208. MR**616373 (82j:12002b)**

Retrieve articles in *Mathematics of Computation*
with MSC:
11R80,
11R04,
11S05

Retrieve articles in all journals with MSC: 11R80, 11R04, 11S05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1984-0736460-5

Article copyright:
© Copyright 1984
American Mathematical Society