For fixed k, the asymptotic Mahler measure of Gaussian periods of conductor kn+1 is n times the Mahler measure of the cyclovariety x0+F_k(x)=0, which is asymptotically (1/2)log k.
On the Mahler measure of $(1+x)(1+y)+z$
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abstract
We prove a conjecture of Boyd and Rodriguez Villegas relating the Mahler measure of the polynomial $(1+x)(1+y)+z$ and the value at $s=3$ of the $L$-function of an elliptic curve of conductor $15$. The proof makes use of the computation by Zudilin and the author of the regulator of certain $K_4$ classes on modular curves.
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The asymptotic Mahler measure of Gaussian periods
For fixed k, the asymptotic Mahler measure of Gaussian periods of conductor kn+1 is n times the Mahler measure of the cyclovariety x0+F_k(x)=0, which is asymptotically (1/2)log k.