Dubickas, Artūras The mean values of logarithms of algebraic integers. (English) Zbl 0923.11145 J. Théor. Nombres Bordx. 10, No. 2, 301-313 (1998). Let \(p >1\). For an algebraic integer \(\alpha\) of degree \(d\) which is not a root of unity, let \(M_p(\alpha)\) be the \(p\)-th root of the mean value \[ {1\over d}\sum_{i=1}^d\bigl| \log| \alpha_i| \bigr| ^p, \] \(\alpha_1=\alpha,\dots,\alpha_d\) being the conjugates of \(\alpha\). The author obtains lower bounds for \(M_p(\alpha)\) of the form \[ M_p(\alpha)>{1\over d}(b_p-\epsilon)\delta(d), \] valid for \(d>d(\epsilon)\), with \(b_p\) being an explicit, but rather complicated constant, and \(\delta(d)=(\log\log d/\log d)^3\). A lower bound is also given for the \(p\)-th root of the mean value of \(p\)-th powers of absolute values of conjugates of \(\alpha\). Reviewer: W.Narkiewicz (Wrocław) MSC: 11R04 Algebraic numbers; rings of algebraic integers Keywords:Mahler’s measure; logarithms of algebraic integers; mean values of logarithms PDFBibTeX XMLCite \textit{A. Dubickas}, J. Théor. Nombres Bordx. 10, No. 2, 301--313 (1998; Zbl 0923.11145) Full Text: DOI Numdam EuDML EMIS References: [1] Bertrand, D., Duality on tori and multiplicative dependence relations. J. Austral. Math. Soc. (to appear). · Zbl 0886.11035 [2] Blanksby, P.E., Montgomery, H.L., Algebraic integers near the unit circle. Acta Arith.18 (1971), 355-369. · Zbl 0221.12003 [3] Cantor, D.C., Straus, E.G., On a conjecture of D.H.Lehmer. Acta Arith.42 (1982), 97-100. · Zbl 0504.12002 [4] Dobrowolski, E., On a question of Lehmer and the number of irreducibile factors of a polynomial. Acta Arith.34 (1979), 391-401. · Zbl 0416.12001 [5] Dubickas, A., On a conjecture of Schinzel and Zassenhaus. Acta Arith.63 (1993), 15-20. · Zbl 0777.11039 [6] Dubickas, A., On the average difference between two conjugates of an algebraic number. Liet. Matem. Rink.35 (1995), 415-420. · Zbl 0862.11058 [7] Langevin, M., Solution des problèmes de Favard. Ann. Inst. Fourier38 (1988), no. 2, 1-10. · Zbl 0634.12002 [8] Lehmer, D.H., Factorization of certain cyclotomic functions. Ann. of Math.34 (1933), 461-479. · JFM 59.0933.03 [9] Louboutin, R., Sur la mesure de Mahler d’un nombre algébrique. C.R.Acad. Sci. Paris296 (1983), 707-708. · Zbl 0557.12001 [10] Matveev, E.M., A connection between Mahler measure and the discriminant of algebraic numbers. Matem. Zametki59 (1996), 415-420 (in Russian). · Zbl 0879.11056 [11] Meyer, M., 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] Mignotte, M., Waldschmidt, M., On algebraic numbers of small height: linear forms in one logarithm. J. Number Theory47 (1994), 43-62. · Zbl 0801.11033 [13] Schinzel, A., Zassenhaus, H., A refinement of two theorems of Kronecker. Michigan Math. J.12 (1965), 81-85. · Zbl 0128.03402 [14] Schur, I., Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Zeitschrift1 (1918), 377-402. · JFM 46.0128.03 [15] Siegel, C.L., The trace of totally positive and real algebraic integers. Ann. of Math.46 (1945), 302-312. · Zbl 0063.07009 [16] Smyth, C.J., On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc.3 (1971), 169-175. · Zbl 0235.12003 [17] Smyth, C.J., The mean values of totally real algebraic integers. Math. Comp.42 (1984), 663-681. · Zbl 0536.12006 [18] Stewart, C.L., Algebraic integers whose conjugates lie near the unit circle. Bull. Soc. Math. France106 (1978), 169-176. · Zbl 0396.12002 [19] Voutier, P., An effective lower bound for the height of algebraic numbers. Acta Arith.74 (1996), 81-95. · Zbl 0838.11065 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.