Mahler measure and elliptic curve \(L\)-functions at \(s = 3\)
Let \(P_k(X,Y)= X+1/X +Y +1/Y +k\). In 1997 \textit{C. Deninger} [J. Am. Math. Soc. 10, No. 2, 259--281 (1997; Zbl 0913.11027)] presented arguments to the assertion that the logarithmic Mahler measure \(m(P_1)\) of \(P_1(X,Y)\) is equal to a rational multiple of the derivative of the \(L\)-function of an elliptic curve at \(s=0\). Later \textit{D. W. Boyd} [Exp. Math. 7, No. 1, 37--82 (1998; Zbl 0932.11069)] found numerically \(m(P_k)\) for some other values of \(k\) which supported the conjecture that a similar situation occurs also for other values of integers \(k\neq0,4\). Deninger's assertion has been confirmed by \textit{M. Rogers} and \textit{W. Zudilin} [Compos. Math. 148, No. 2, 385--414 (2012; Zbl 1260.11062)], and Boyd's conjecture has been confirmed in particular cases by \textit{M. Rogers} and \textit{W. Zudilin} [\(k=8\); Int. Math. Res. Not. 2014, No. 9, 2305--2326 (2014; Zbl 1378.11091)] the author [\(k=5\), Int. J. Number Theory 6, No. 3, 705--711 (2010; Zbl 1201.11098)] and the author and \textit{M. D. Rogers} [\(k=2\); Algebra Number Theory 1, No. 1, 87--117 (2007; Zbl 1172.11037)]. In this paper, the author shows that a similar situation occurs also for other polynomials, gives a formula for \(m(Z+(X+1)(Y+1))\) and shows that for a class of polynomials in three variables the calculation of the Mahler measure can be reduced to the case of a two-variable polynomial.
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