Apéry limits and Mahler measures

Identities between Mahler measures

The purpose of this short note is to give a proof of the following identity between (logarithmic) Mahler measures m(y^2+2xy+y-x^3-2x^2-x)=5/7*m(y^2+4xy+y-x^3+x^2) which is one of many examples that arise from the comparison of Mahler measures and special values of L-functions.

Wandering points for the Mahler measure

Mahler's measure defines a dynamical system on the algebraic numbers. In this paper, we study the problem of which number fields have points which wander under the iteration of Mahler's measure. We completely solve the problem for all abelian number fields, and more generally, for all extensions of the rationals of degree at most five.

Functional equations for Mahler measures of genus-one curves

In this paper we will establish functional equations for Mahler measures of families of genus-one two-variable polynomials. These families were previously studied by Beauville, and their Mahler measures were considered by Boyd, Rodriguez-Villegas, Bertin, Zagier, and Stienstra. Our functional equations allow us to prove identities between Mahler measures that were conjectured by Boyd. As a corollary, we also establish some new transformations for hypergeometric functions.

Higher Mahler measure for cyclotomic polynomials and Lehmer's question

The $k$-higher Mahler measure of a nonzero polynomial $P$ is the integral of $\log^k|P|$ on the unit circle. In this note, we consider Lehmer's question (which is a long-standing open problem for $k=1$) for $k>1$ and find some interesting formulae for 2- and 3-higher Mahler measure of cyclotomic polynomials.

四元数马勒测度

Quaternionic Mahler measure

引入非交换多项式的四元数马勒测度，扩展经典复马勒测度。建立单变量和双变量切片正则多项式的四元数马勒测度的存在性，研究实及切片正则多项式的四元数马勒测度并考虑相关莱默问题，证明多种公式。

The infimum in the metric Mahler measure

Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number $α$ by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved as well as establish its analog for the ultrametric Mahler measure.

A study of inverse trigonometric integrals associated with three-variable Mahler measures, and some related identities

We prove several identities relating three-variable Mahler measures to integrals of inverse trigonometric functions. After deriving closed forms for most of these integrals, we obtain ten explicit formulas for three-variable Mahler measures. Several of these results generalize formulas due to Condon and Lalín. As a corollary, we also obtain three $q$-series expansions for the dilogarithm.

Mahler Measure of "Almost" Reciprocal Polynomials

Here we give a lower bound of the Mahler measure on a set of polynomials that are "almost" reciprocal. Here "almost" reciprocal means that the outermost coefficients of each polynomial mirror each other in proportion, while this pattern breaks down for the innermost coefficients.

四次伽罗瓦数域中的最小马勒测度

Minimal Mahler Measure in Quartic Galois Number Fields

本研究聚焦四次伽罗瓦数域中最小整马勒测度与域判别式的关联，推导了有条件依赖ABC猜想的密度结论及若干无条件结论。

Hypergeometric formulas for lattice sums and Mahler measures

We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd's conjectured identities between Mahler measures and special values of $L$-series of elliptic curves.

Higher Mahler measures and zeta 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.

Some examples of Mahler measures as multiple polylogarithms

The Mahler measures of certain polynomials of up to five variables are given in terms of multiple polylogarithms. Each formula is homogeneous and its weight coincides with the number of variables of the corresponding polynomial.

Asymptotic nature of higher Mahler measure

We consider Akatsuka's zeta Mahler measure as a generating function of higher Mahler measure $m_k(P)$ of a polynomial $P,$ where $m_k(P)$ is the integral of $\log^{k}\left| P \right|$ over the complex unit circle. Restricting ourselves to $\displaystyle P(x)=x+r$ with $|r|=1$ we show some new asymptotic results regarding $m_k(P)$, especially $\left| m_k(P) \right|/k! \to 1/π$ as $k \to \infty.$

The Mahler measure of algebraic numbers: a survey

A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (`house') of an algebraic integer are also discussed. Some generalisations are given too, though not to Mahler measure of polynomials in more than one variable.

Mahler measure numerology

We discuss some (conjectural) evaluations of $L$-values attached to elliptic curves of conductors 15, 21, 24 and 32 as "hypergeometric periods". These numerical observations are motivated by the Mahler measures of three-variable polynomials.

Linear Mahler Measures and Double L-values of Modular Forms

We consider the Mahler measure of the polynomial 1+x_1+x_2+x_3+x_4, which is the first case not yet evaluated explicitly. A conjecture due to F. Rodriguez-Villegas represents this Mahler measure as a special value at the point 4 of the L-function of a modular modular form of weight 3. We prove that this Mahler measure is equal to a linear combination of double L-values of certain meromorphic modular forms of weight 4.

New $_5F_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1/π$

New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the $_5F_4$ hypergeometric function. We use these results to obtain two explicit $_5F_4$ evaluations, and several new formulas for $1/π$.

An areal analog of Mahler's measure

We consider a version of height on polynomial spaces defined by the integral over the normalized area measure on the unit disk. This natural analog of Mahler's measure arises in connection with extremal problems for Bergman spaces. It inherits many nice properties such as the multiplicative one. However, this height is a lower bound for Mahler's measure, and it can be substantially lower. We discuss some similarities and differences between the two.

A generalization of a theorem of Boyd and Lawton

The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of $\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of $\log|P|$ for possibly different $P$'s), multiple Mahler measure (involving products of $\log|P|$ for possibly different $P$'s), and higher Mahler measure (involving $\log^k|P|$).

Mahler 测度的一个不变性质

An invariant property of Mahler measure

本文提出保持多项式 Mahler 测度的变量替换，用以构造具有指定 Mahler 测度的非平凡多项式，并证明 Boyd 与 Brunault 的若干数值猜想。

Limiting value of higher Mahler measure

We consider the $k$-higher Mahler measure $m_k(P)$ of a Laurent polynomial $P$ as the integral of $\log ^k \left| P \right|$ over the complex unit circle. In this paper we derive an explicit formula for the value of $\left| m_k(P) \right|/k!$ as $k \to \infty.$

Mahler measure of polynomials defining genus 2 and 3 curves

In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of $L$-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these famili…

Lower bounds for Mahler measure that depend on the number of monomials

We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients, and the number of monomials. In one variable our result generalizes a classical inequality of Mahler. In $M$ variables our result depends on $\mathbb{Z}^M$ as an ordered group, and in general our lower bound depends on the choice of ordering.

On the Mahler measure of $1+X+1/X+Y+1/Y$

We prove a conjectured formula relating the Mahler measure of the Laurent polynomial $1+X+X^{-1}+Y+Y^{-1}$, to the $L$-series of a conductor 15 elliptic curve.

A shifted Mahler measure identity for Boyd's family

Recently the second author and Qin numerically verified some Mahler measure identities of genus 2 and 3 polynomial families. In this paper, we use the elliptic regulator to prove an identity invoving shifted Mahler measure for Boyd's family.

The Mahler measure and the L-series of a singular K3-surface

We present the first example of a polynomial defining a singular K3-surface whose Mahler measure is expressed in terms of the Mahler measure of the faces of its Newton polyhedron and the L-series of the K3-surface.

多变量多项式的面积 Mahler 测度的求值

Evaluations of the areal Mahler measure of multivariable polynomials

本文研究了多变量多项式的面积 Mahler 测度，通过在单位圆盘乘积上积分进行求值，并发现了产生 $L$-函数特殊值的例子。

一族非回火多项式的马勒测度与博伊德猜想

Mahler measures of a family of non-tempered polynomials and Boyd's conjectures

本文通过建立非回火多项式与回火多项式马勒测度之间的新恒等式，结合 Brunault-Mellit-Zudilin 公式及附加泛函恒等式，证明了导子为 30 的椭圆曲线的博伊德猜想。

Minimal Mahler Measure in Cubic Number Fields

The minimal integral Mahler measure of a number field $K$, $M(\mathcal{O}_K)$, is the minimal Mahler measure of a non-torsion primitive element of $\mathcal{O}_K$. Upper and lower bounds, which depend on the discriminant, are known. We show that for cubics, the lower bounds are sharp with respect to its growth as a function of discriminant. We construct an algorithm to compute $M(\mathcal{O}_K)$ for all cubics with absolute value of the discriminant bounded by $N$.

Mahler measure and the WZ algorithm

We use the Wilf-Zeilberger method to prove identities between Mahler measures of polynomials. In particular, we offer a new proof of a formula due to Lalín, and we show how to translate the identity into a formula involving elliptic dilogarithms. This work settles a challenge problem proposed by Kontsevich and Zagier in their paper "Periods".

Mahler measure of some n-variable polynomial families

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.

Mahler Measure of 3D Landau-Ginzburg Potentials

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.

Operator analogues of Mahler's measure

Motivated by a geometric meaning of Mahler's measure, we introduce two operator analogues of Mahler's measure. This leads to some interesting equalities and inequalities between the two operator-theoretic Mahler measures and the classical Mahler measure. In order to apply these results to the operator version of Lehmer's problem, we introduce and study an important class of operators, the so-called subharmonic operators. It is shown that the operator version of Lehmer's problem fails under some mild condition.

Mahler measure under variations of the base group

We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lueck-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In particular, we study how to obtain the Mahler measure over an infinite group as limit of Mahler measures over finite groups, for example, in the classical case of the free abelian group or the infinite dihedral group, and others.

Mahler measures and computations with regulators

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]

Mahler Measure Variations, Eisenstein Series and Instanton Expansions

This paper points at an intriguing inverse function relation between Eisenstein series connected with ``Modular Mahler Measures'' and instanton numbers for ``Non-Critical Strings''. In a companion paper Mahler measures are related to dimer models.

Mahler measure of some singular K3-surfaces

We study the Mahler measure of the three-variable Laurent polynomial x + 1/x + y + 1/y + z + 1/z - k where k is a parameter. The zeros of this polynomial define (after desingularization) a family of K3-surfaces. In favorable cases, the K3-surface has Picard number 20, and the Mahler measure is related to its L-function. This was first studied by Marie-Jose Bertin. In this work, we prove several new formulas, extending the earlier work of Bertin.

Norms extremal with respect to the Mahler measure

In a previous paper, the authors introduced several vector space norms on the space of algebraic numbers modulo torsion which corresponded to the Mahler measure on a certain class of numbers and allowed the authors to formulate L^p Lehmer conjectures which were equivalent to their classical counterparts. In this paper, we introduce and study several analogous norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We evaluate these norms on ce…

Minimal Mahler measure in real quadratic fields

We consider upper and lower bounds on the minimal height of an irrational number lying in a particular real quadratic field.

The Mahler measure of a Calabi-Yau threefold and special L-values

The aim of this paper is to prove a Mahler measure formula of a four-variable Laurent polynomial whose zero locus defines a Calabi-Yau threefold. We show that its Mahler measure is a rational linear combination of a special L-value of the normalized newform in S_4(Gamma_0(8)) and a Riemann zeta value. This is equivalent to a new formula for a 6F5-hypergeometric series evaluated at 1.

On Mahler's transcendence measure for $e$

We present a completely explicit transcendence measure for $e$. This is a continuation and an improvement to the works of Borel, Mahler and Hata on the topic. Furthermore, we also prove a transcendence measure for an arbitrary positive integer power of $e$. The results are based on Hermite-Padé approximations and on careful analysis of common factors in the footsteps of Hata.

On the Mahler measure of resultants in small dimensions

We prove that sparse resultants having Mahler measure equal to zero are those whose Newton polytope has dimension one. We then compute the Mahler measure of resultants in dimension two, and examples in dimension three and four. Finally, we show that sparse resultants are tempered polynomials. This property suggests that their Mahler measure may lead to special values of L-functions and polylogarithms.

A Mahler measure of a K3-hypersurface expressed as a Dirichlet L-series

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.

Laurent多项式的广义Mahler测度

Generalized Mahler measures of Laurent polynomials

本文研究二元Laurent多项式在任意环面上的广义Mahler测度，建立其与标准Mahler测度的关系，并对Boyd多项式族显式计算r=4情形，最终推广至多变量。

On linear independence measures of the values of Mahler functions

In this paper, we estimate the linear independence measures for the values of a class Mahler functions of degree one and two. For the purpose, we study the determinants of suitable Hermite-Padé approximation polynomials. Based on the non-vanishing of these determinants, we apply the functional equations to get an infinite sequence of approximations which is used to produce the linear independence measures.

Two inequalities on the areal Mahler measure

Recent work of Pritsker defines and studies an areal version of the Mahler measure. We further explore this function with a particular focus on the case where its value is small, as this is most relevant to Lehmer's conjecture. In this situation, we provide improvements to two inequalities established in Pritsker's original paper.

The elliptic trilogarithm and Mahler measures of $K3$ surfaces

The aim of this paper is to derive explicitly a connection between the Zagier elliptic trilogarithm and Mahler measures of a certain family of three-variable polynomials defining K3 surfaces. In addition, we prove some linear relations satisfied by the elliptic trilogarithm evaluated at torsion points on elliptic curves. This result can be viewed as a higher dimensional analogue of exotic relations of the elliptic dilogarithm.

Mahler measures as linear combinations of $L$-values of multiple modular forms

We study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves $L$-values of at most one newform and/or at most one quadratic character. In this paper, we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to $L$-values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular $L$-values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen…

穿过面积Mahler测度的随机游走：复平面中的步长

Random walks through the areal Mahler measure: steps in the complex plane

本文通过随机游走和Zeta Mahler函数的面积类比，证明了两个多项式族的面积Mahler测度可表示为标准Mahler测度与Deninger循环体积的线性组合。

一类具有任意多变量的多项式的Mahler测度

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。

Mahler measure of $P_d$ polynomials

This article investigates the Mahler measure of a family of 2-variate polynomials, denoted by $P_d, d\geq 1$, unbounded in both degree and genus. By using a closed formula for the Mahler measure introduced in "Volume function and Mahler measure of exact polynomials" (by Guilloux and Marché), we are able to compute $m(P_d)$, for arbitrary $d$, as a sum of the values of dilogarithm at special roots of unity. We prove that $m(P_d)$ converges and the limit is proportional to $ζ(3)$, where $ζ$ is the Riemann zeta function.

A collection of metric Mahler measures

Let $M(α)$ denote the Mahler measure of the algebraic number $α$. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the author used similar techniques to study a non-Archimedean version. We show how to generalize the above constructions in order to associate, to each point in $(0,\infty]$, a metric version $M_x$ of the Mahler measure, each having a triangle inequality of a different strength. We are able to compute $M_x(α)$ for sufficiently small $x$, identifying, in the process, a function…

Mahler measures, K-theory and values of L-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…

Mahler measures of polynomials that are sums of a bounded number of monomials

We study Laurent polynomials in any number of variables that are sums of at most $k$ monomials. We first show that the Mahler measure of such a polynomial is at least $h/2^{k-2}$, where $h$ is the height of the polynomial. Next, restricting to such polynomials having integer coefficients, we show that the set of logarithmic Mahler measures of the elements of this restricted set is a closed subset of the nonnegative real line, with $0$ being an isolated point of the set. In the final section, we discuss the extent to which such an integer polynomial of Mahler measure $1$ is determined by its $k…