Multiple Mahler measures of $1+x+y_j$ and $1+x+y_j+z_j$

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Numbers

$n$

$k$

$M_{n,k}$

0.3230659472194505140936365107238063940722418407805870161308684703610151128072698420837876090893713921

comment: Smyth's three-term logarithmic Mahler measure.

0.1827704518720251596080461296273361321354388779118664930817286921522230522670223193148476556244176339

0.1097971263652280137499847601886304382185305979676779110376504269526428613474566456762761110516364649

0.06812635402406724391249902914787973139322768757636880598875151659092755168278841135441816980590926671

0.04313777375696840048727843669258520532507248460656657395021760106968641666546015687395025130877622128

0.02769919805959168345364699674685258972506855715493536601912279616802708247462605288984228029678868286

0.01796847619389192719495973082189465777461964727355937723019764891118969162818479835753405853383742967

0.01174736662727979744151140949469509729987436112011018721658601003946199267786833489192157186476547031

0.007727514521987666121627107608321668400787337025429054140377171021299911587520920450719191082410943021

10:

0.005108572434902816684225620589377376658701114812353081425385794297999801194005732229853548905743721860

11:

0.003391123479776083221584555526057218567432779898385605596285774543645304613906563949606366610631432672

12:

0.002258844923102715255305863258666156753498296489564927854847388940412430778546128428462719548480203375

0.4262783988175057909235214265961668730580067696296351075416064580265294512291165814891241883322429436

comment: Smyth's four-term logarithmic Mahler measure.

0.2274871670663211118351857116685368739313129907527546276159811317399916635615715892693614251380466792

0.1320653038216695182572393174558585431486392073214072234818037016025013663067605720866639000278648885

0.08009383817867761123079405076484346461861010532971502292291438563072605448688029451459713017180401461

0.04988566338579497737792690261497906505080832050071562398851698952604073945775970438548729560655085797

0.03163148603387023753180012122564402336978808529562818428344892564644630126565361736176470172929712074

0.02031564652409777745353650501918629348488631444579297477059643492822973312637322633047703802871137715

0.01317417359243074409049219187672044520661726276629877449576440452636668216403716653235175820453275736

0.008607359376207402004112926706951741232436580303593016510159949994565823632368980634705118835299576076

10:

0.005657428496993149111629512084006994769672815878463972971727491048784894954794185415851707682155455944

11:

0.003736770676693703237227653153378708256134658757242187402248426177280763869666096211431267028096034550

12:

0.002478255131913061090300252305950474178769627550655319450479466698410173945483208399581465139516085127

Definition

For $P_{n,j}=1+x+\sum_{\ell=1}^{n-2}y_{j,\ell}$, the multiple Mahler measure is $M_{n,k}=\mathbb{E}\prod_{j=1}^k\log|P_{n,j}|$, with independent Haar-uniform unit-circle coordinates and the same $x$ in every factor [1].

Parameters

$n$

— number of summands ($n\geq3$)

$k$

— number of logarithmic factors ($k\geq1$)

Formulas

(1)

$M_{3,k}=\dfrac1\pi\int_{\pi/3}^{\pi}\log^k\!\left(2\sin\dfrac{\theta}{2}\right)\,d\theta=\dfrac23\int_0^1\log^k\!\left(2\cos\dfrac{\pi t}{3}\right)\,dt$ for $k\geq1$, by Jensen's formula [1], Theorem 2.1.

(2)

$M_{4,k}=\int_0^1g(t)^k\,dt$, where $g(t)=t\log(2\sin(\pi t/2))+\operatorname{Cl}_2(\pi t)/\pi$ for $0<t\leq1$ and $g(0)=0$ by continuity [1], Theorem 3.3.

(3)

$g(t)=t\left(1-2\sum_{m=1}^{\infty}\dfrac{\zeta(2m)}{2m+1}\left(\dfrac t2\right)^{2m}\right)$ for $0\leq t\leq1$. This follows from $g^{\prime}(t)=(\pi t/2)\cot(\pi t/2)$ and the cotangent partial fractions [3]; $\zeta$ is the Riemann zeta function.

(4)

$M_{3,1}=\operatorname{Cl}_2(\pi/3)/\pi$, $M_{3,2}=\pi^2/54$, and $M_{3,3}=9\operatorname{Cl}_4(\pi/3)/(2\pi)-\zeta(3)$ [1], Example 3.1. The Clausen factors are $\operatorname{Cl}_2(\pi/3)$ and $\operatorname{Cl}_4(\pi/3)$, and $\zeta(3)$ is Apéry's constant.

(5)

$M_{4,1}=7\zeta(3)/(2\pi^2)$ and $M_{4,2}=\dfrac{8\operatorname{Li}_4(1/2)+7\log(2)\zeta(3)+\log^4(2)/3}{\pi^2}-\dfrac{\log^2(2)}3-\dfrac{23\pi^2}{360}$ [1], Example 3.4 with equations (10) and (24). The polylogarithm value is $\operatorname{Li}_4(1/2)$.

Comments

(6)

For $n=3$ write $y_{j,1}=y_j$, so $P_{3,j}=1+x+y_j$. For $n=4$ write $y_{j,1}=y_j$ and $y_{j,2}=z_j$, so $P_{4,j}=1+x+y_j+z_j$. The variables $y_j,z_j$ are private to factor $j$.

(7)

Each circle has total measure one, and the logarithm is natural. These are raw joint logarithmic moments. The absolute value encloses each polynomial, not its logarithm, and there is no factorial divisor, centering, standardization or exponentiation. The integrals at zeros are convergent improper integrals. The notation $\mu_k(1+x+y_*)$ and $\mu_k(1+x+y_*+z_*)$ in [2] denotes these two sharing patterns.

(8)

When $k=1$, $M_{n,1}$ is the ordinary logarithmic Mahler measure of $1+x_1+\cdots+x_{n-1}$. The ordinary-measure companion contains this first-order family; its entries $n=3$ and $n=4$ are included here as the first moments.

(9)

Identifying every $P_{n,j}$ with a single polynomial would instead give a higher Mahler measure, $\mathbb{E}(\log|P|)^k$. In the cyclotomic and geometric-sum higher-measure tables, all monomials are powers of one phase. Their phase dependencies differ from the shared and private coordinates used here.

(10)

Write $\operatorname{Cl}_{2r}(\theta)=\sum_{m=1}^{\infty}\sin(m\theta)/m^{2r}$ for the even-order Clausen functions. That companion uses $t=\theta/\pi$, so $\theta=\pi/3$ corresponds to its parameter $t=1/3$.

Programs

(P1)

Sage

# Download https://numberdb.org/files/T685/generate.py?raw=1 as generate.py.
# Install numberdb with sage -pip install numberdb.
from generate import moment
moment(4, 2, digits=100)

References

[1]

Jonathan M. Borwein and Armin Straub, Log-sine evaluations of Mahler measures, Journal of the Australian Mathematical Society 92 (2012), 15–36. Theorem 2.1, Theorem 3.3 and Examples 3.1 and 3.4. (arXiv) (doi)

Links

[2]

Borwein and Straub, author preprint

[3]

DLMF 4.22.3: partial fractions for cotangent

Similar tables

Mahler measures of $1+x_1+\dots+x_{n-1}$ (uniform random walks in the plane) — Ordinary logarithmic Mahler measures of the same linear forms, equal to the k=1 boundary.

Values of the Clausen functions $\mathrm{Cl}_s$ at rational multiples of $\pi$ — Clausen values in the low-order formulas and the four-term integral reduction.

Values of the Riemann zeta function at rational numbers — Zeta values in the conditional-mean series and low-order formulas.

Values of the polylogarithm $\mathrm{Li}_s$ at rational arguments — The polylogarithm value Li_4(1/2) in the four-term second moment.

Higher Mahler measure of cyclotomic polynomials — Higher logarithmic moments of cyclotomic polynomials, with repeated logarithms of one polynomial.

Higher Mahler measures of $(x^n-1)/(x-1)$ — Higher logarithmic moments of geometric sums whose monomials depend on one phase.

Higher Mahler measures of $1+x_1+...+x_{n-1}$ — The higher moments $H_{n,k}=\mathbb{E}(\log|L_n|)^k$ repeat one polynomial in every logarithmic factor. The multiple moments $M_{n,k}$ here share only the phase $x$, with independent private phases in each factor. At first order both constructions reduce to the ordinary logarithmic Mahler measure; higher orders measure different phase dependencies.

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it holds $n=3,4$ and every $1\leq k\leq12$, covering both sharing patterns in Borwein and Straub and both parities of the factor count. The noncanonical order cutoff includes the first-order controls and extends beyond the low-order closed forms while allowing certified 100-digit computation and independent checks of every entry. Higher summand counts require additional integral reductions; the attached argument-selection.md explains the bounds)

How they were obtained:

The three-term values use certified Arb integration of the compact log-cosine integral. The four-term values use a polynomial truncation of the convergent series for the conditional mean g, with a proved uniform tail bound propagated through its kth power. Real and complex ball arithmetic enclose argument rounding and every arithmetic operation. The 100-digit computation uses 445 working bits and 235 series terms; each result must retain at least 373 accurate bits. The minimum measured relative accuracy was 431 bits, about 129 decimal digits. The attached certification.md derives the bounds, and numerical-evidence.json records the actual radii.

All 24 entries are recomputed at 140 requested digits and compared with independent mpmath quadrature of the log-sine and Clausen forms at 75 and 125 decimal working digits. Known integral controls are run first. Five low-order formulas from Borwein and Straub provide additional checks using pi, zeta, Clausen and polylogarithm values. Positivity, the bound by log(2) between successive moments, and log-convexity are checked over every applicable order. The numerical quadrature is an independent agreement check; the ball integration and explicit series remainder justify the certified precision.