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Báo cáo toán học: "A natural series for the natural logarithm"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A natural series for the natural logarithm. | A natural series for the natural logarithm Oliver T. Dasbach kasten@math.lsu.edu Louisiana State University Department of Mathematics Baton Rouge LA 70803 http www.math.lsu.edu kasten Submitted Feb 22 2007 Accepted Feb 27 2008 Published Mar 7 2008 Mathematics Subject Classification 57M25 57M50 40A05 05A10 Abstract Rodriguez Villegas expressed the Mahler measure of a polynomial in terms of an infinite series. Luck s combinatorial L2-torsion leads to similar series expressions for the Gromov norm of a knot complement. In this note we show that those formulae yield interesting power series expansions for the logarithm function. This generalizes an infinite series of Lehmer for the natural logarithm of 4. 1 The abelian case the Mahler measure For a Laurent polynomial P in the group ring C Zr let the conjugate P be defined by sending every g 2 Zr to g 1 and every coefficient ag to its complex conjugate ag. The logarithmic Mahler measure see e.g. EW99 of P is given by m P Ị J0 ln P u2- .w - dti dtr. The following theorem is due to Rodriguez Villegas RV99 . Independently it also appears in the study of the combinatorial L2-torsion due to Luck. Further discussions are given in DL08 . We include a proof along the lines of RV99 for completeness. Theorem 1.1 RV99 see also Luc02 Den06 . For k greater than the l1 -norm of the coefficients of P we have 2m P m PP 2ln k - X 1 IỴ1 - APPẠ 1 n k2 I n n 1 L x -I u 1 where P 0 denotes the constant coefficient of the Laurent polynomial P. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N5 1 Proof. Since PP k2 1 - 1 - -1 PP k2 we have m PP 2 ln k m 1 Set Q 1 - 712PP and let k2 Q x L 1 .Q- .e-r di1 dt X xn Q e2mti e2 ir ndti dir n o Jo Jo X xn Qn o n o The choice of k ensures convergence. Now m 1 xQ In 1 - Q e2tó1 ln 1 - Q e Jo Jo - i u Q z - 1 dz Jo z e2 ir dti dtr e2mtr dti --dir X 1 xn Qlo __ n n 1 setting x 1 yields the result. 2 A power series for the natural logarithm Here we study an application of Equation 1 which leads to an .