The mean values of logarithms of algebraic integers

Artūras Dubickas

Artūras Dubickas

Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 301-313

Abstract (VO)
Résumé

Let $α$ be an algebraic integer of degree $d$ with conjugates $α_{1} = α, α_{2}, \dots, α_{d}$ . In the paper we give a lower bound for the mean value

M_{p} (α) = \sqrt[p]{\frac{1}{d} \sum_{i = 1}^{d} | log | α_{i} {| |}^{p}}

when

α

is not a root of unity and

p > 1

.

Soit $α_{1} = α, α_{2}, \dots, α_{d}$ l’ensemble des conjugués d’un entier algébrique $α$ de degré $d$ , n’étant pas une racine de l’unité. Dans cet article on propose de minorer

M_{p} (α) = \sqrt[p]{\frac{1}{d} \sum_{i = 1}^{d} | log | α_{i} {| |}^{p}}

où

p > 1

.

MR Zbl

@article{JTNB_1998__10_2_301_0,
     author = {Art\={u}ras Dubickas},
     title = {The mean values of logarithms of algebraic integers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {301--313},
     year = {1998},
     publisher = {Universit\'e Bordeaux I},
     volume = {10},
     number = {2},
     zbl = {0923.11145},
     mrnumber = {1828247},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_1998__10_2_301_0/}
}

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Artūras Dubickas. The mean values of logarithms of algebraic integers. Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 301-313. https://jtnb.centre-mersenne.org/item/JTNB_1998__10_2_301_0/

Bibliographie
Cité par

[1] D. Bertrand, Duality on tori and multiplicative dependence relations. J. Austral. Math. Soc. (to appear). | Zbl | MR

[2] P.E. Blanksby, H.L. Montgomery, Algebraic integers near the unit circle. Acta Arith. 18 (1971), 355-369. | Zbl | MR | EuDML

[3] D.C. Cantor, E.G. Straus, On a conjecture of D.H.Lehmer. Acta Arith. 42 (1982), 97-100. | Zbl | MR | EuDML

[4] E. Dobrowolski, On a question of Lehmer and the number of irreducibile factors of a polynomial. Acta Arith. 34 (1979), 391-401. | Zbl | MR | EuDML

[5] A. Dubickas, On a conjecture of Schinzel and Zassenhaus. Acta Arith. 63 (1993), 15-20. | Zbl | MR | EuDML

[6] A. Dubickas, On the average difference between two conjugates of an algebraic number. Liet. Matem. Rink. 35 (1995), 415-420. | Zbl | MR

[7] M. Langevin, Solution des problèmes de Favard. Ann. Inst. Fourier 38 (1988), no. 2, 1-10. | Numdam | Zbl | MR | EuDML

[8] D.H. Lehmer, Factorization of certain cyclotomic functions. Ann. of Math. 34 (1933), 461-479. | Zbl | MR | JFM

[9] R. Louboutin, Sur la mesure de Mahler d'un nombre algébrique. C.R.Acad. Sci. Paris 296 (1983), 707-708. | Zbl | MR

[10] E.M. Matveev, A connection between Mahler measure and the discriminant of algebraic numbers. Matem. Zametki 59 (1996), 415-420 (in Russian). | Zbl | MR

[11] M. Meyer, Le problème de Lehmer: méthode de Dobrowolski et lemme de Siegel "à la Bombieri-Vaaler". Publ. Math. Univ. P. et M. Curie (Paris VI), 90, Problèmes Diophantiens (1988-89), No.5, 15 p.

[12] M. Mignotte, M. Waldschmidt, On algebraic numbers of small height: linear forms in one logarithm. J. Number Theory 47 (1994), 43-62. | Zbl | MR

[13] A. Schinzel, H. Zassenhaus, A refinement of two theorems of Kronecker. Michigan Math. J. 12 (1965), 81-85. | Zbl | MR

[14] I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Zeitschrift 1 (1918), 377-402. | MR | JFM

[15] C.L. Siegel, The trace of totally positive and real algebraic integers. Ann. of Math. 46 (1945), 302-312. | Zbl | MR

[16] C.J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc. 3 (1971), 169-175. | Zbl | MR

[17] C.J. Smyth, The mean values of totally real algebraic integers. Math. Comp. 42 (1984), 663-681. | Zbl | MR

[18] C.L. Stewart, Algebraic integers whose conjugates lie near the unit circle. Bull. Soc. Math. France 106 (1978), 169-176. | Zbl | MR | Numdam

[19] P. Voutier, An effective lower bound for the height of algebraic numbers. Acta Arith. 74 (1996), 81-95. | Zbl | MR