Marie Jose Bertin
We present another example of a 3-variable polynomial defining a K3-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet L-series.
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We express the logarithmic Mahler measure of two families of polynomials defining K3 hypersurfaces of generic Picard number 19 in terms of Eisenstein-Kronecker series. For some singular K3 surfaces (i.e of Picard number 20) we write this measure as Hecke L-series of weight 3 with "Gr\" ossencharacters".
The arithmetic volume of hypersurfaces in toric varieties and Mahler measures
该研究确定了光滑射影环面簇中超曲面的典范算术体积，并由此证明了这类超曲面上的广义Hodge指标定理，为算术代数几何领域相关问题提供了新结果。
Mahler measure, motivic regulators and Dirichlet $L$-values
本文通过研究动机上同调和狄利克雷L函数导数的关系，建立了马勒测度与调节器之间的公式，并在特定假设下推导了单个L值的恒等式。
In this paper we prove that the Mahler measures of the Laurent polynomials $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k$, $(x+x^{-1})^2(y+y^{-1})^2(1+z)^3z^{-2}-k$, and $x^4+y^4+z^4+1+k^{1/4}xyz$, for various values of $k$, are of the form $r_1 L'(f,0)+r_2 L'(χ,-1)$, where $r_1,r_2\in \mathbb{Q}$, $f$ is a CM newform of weight 3, and $χ$ is a quadratic character. Since it has been proved that these Maher measures can also be expressed in terms of logarithms and $_5F_4$-hypergeometric series, we obtain several new hypergeometric evaluations and transformations from these results.
Relating Mahler measures and Dirichlet $L$-values: new evidence for Chinburg's conjectures
本文通过新的数值例子验证了陈伯格猜想，展示了8个强形式和18个弱形式的实例，基于Boyd和Rodriguez-Villegas的工作，证明了允许环面系数的弱猜想。
The Mahler measures of some n-variable polynomial families are given in terms of special values of the Riemann zeta function and a Dirichlet L-series, generalizing the results of \cite{L}. The technique introduced in this work also motivates certain identities among Bernoulli numbers and symmetric functions.
We consider a generalization of the Mahler measure of a multivariable polynomial $P$ as the integral of $\log^k|P|$ in the unit torus, as opposed to the classical definition with the integral of $\log|P|$. A zeta Mahler measure, involving the integral of $|P|^s$, is also considered. Specific examples are computed, yielding special values of zeta functions, Dirichlet $L$-functions, and polylogarithms.
We express the Mahler measures of $23$ families of Laurent polynomials in terms of Eisenstein-Kronecker series. These Laurent polynomials arise as Landau-Ginzburg potentials on Fano $3$-folds, $16$ of which define $K3$ hypersurfaces of generic Picard rank $19$, and the rest are of generic Picard rank $< 19$. We relate the Mahler measure at each rational singular moduli to the value at $3$ of the $L$-function of some weight-$3$ newform. Moreover, we find $10$ exotic relations among the Mahler measures of these families.
The Mahler measure of a family of polynomials with arbitrarily many variables
本研究针对任意多变量的无限多项式族，通过积分变换与迭代化简为六次单位根处多重对数求值，推导出其Mahler测度的精确公式，得到Riemann zeta函数与导子3的Dirichlet L函数相关值。
In this work we apply the techniques that were developed in [Lalin: An algebraic integration for Mahler measure] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of the Riemann zeta function or Dirichlet L-series. The examples may be understood in terms of evaluations of regulators. Moreover, we apply the same techniques to the computation of generalized Mahler measures, in the sense of Gon and Oyanagi [Gon, Oyanagi: Generalized Mahler measures and multiple sine functions]
The Mahler measure of a polynomial $P$ in $n$ variables is defined as the mean of $\log|P|$ over the $n$-dimensional torus. For certain polynomials with integer coefficients in two variables the Mahler measure is known to be related to special values of L-functions of arithmetic objects (e.g. Dirichlet characters and elliptic curves over $\mathbb{Q}$). Inspired by work of Deninger Boyd has investigated this relationship numerically. In this paper we reduce some conjectures of Boyd to Beilinson`s conjectures on special values of L-functions. The methods in use are widely of K-theoretical nature…
The Mahler measure of exact polynomials in three variables
研究通过椭圆曲线L值和Bloch-Wigner双曲值表达三变量多项式马尔可夫度量，基于Beilinson猜想，并推广Lalín的结果，应用于Boyd和Brunault的猜想。
An exact family of bivariate polynomials and Variants of Chinburg's Conjectures
本文通过研究二元多项式族$P_d(x,y)$的Mahler测度，将其表示为狄利克雷$L$-函数的线性组合，从而为Chinburg猜想及其推广版本提供了解答，覆盖多个导子情况。
The weak Chinburg conjecture on Mahler measures
本文通过构造有理多项式 $R_f$ 并证明其 Mahler 测度等于 $4wL'(\chi_{-f},-1)$，从而证明了弱二元 Chinburg 猜想。
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