Higher Mahler measures of $1+x_1+...+x_{n-1}$

—   number of summands ($n\geq2$)

—   logarithmic moment order ($k\geq2$)

Let $W_n(s)=\mathbb{E}|L_n|^s$, the radial moment function. For $s$ in a neighborhood of $0$, $H_{n,k}=W_n^{(k)}(0)$. In particular, the coefficient of $s^k$ in the Taylor series of $W_n$ is $H_{n,k}/k!$ [1], section 4.

If $p_n(x)$ is the probability density of the radial distance after $n$ independent unit steps, then $H_{n,k}=\int_0^n p_n(x)\log^k(x)\,dx$ [2], equations (3), (4) and (7).

$H_{4,2}=\dfrac{24\operatorname{Li}_4(1/2)-18\zeta(4)+21\zeta(3)\log2-6\zeta(2)\log^2 2+\log^4 2}{\pi^2}$ [2], equation (70). The factors $\operatorname{Li}_4(1/2)$, $\zeta(2)$, $\zeta(3)$ and $\zeta(4)$ have their own entries.

Each circle has total measure one, every coefficient of $L_n$ is $1$, and the logarithm is natural. These are signed raw logarithmic moments, with no factorial divisor, centering, standardization or exponentiation. The absolute value encloses the polynomial, not its logarithm. Integrals at zeros are convergent improper integrals.

The integer $n$ counts all summands, including the constant $1$, so $L_n$ has $n-1$ independent variables. Rotational invariance identifies $|L_n|$ with the distance from the origin after $n$ independent unit steps in the plane [2].

The same integral at $k=1$ is the ordinary logarithmic Mahler measure. The present table begins at $k=2$; its values are moments of the logarithm, rather than the radial power moments in Moments of uniform random walks in the plane.

Multiple Mahler measures with shared and private phases integrate products of logarithms of different polynomials. Here every logarithmic factor has the same argument. The cyclotomic and geometric-sum higher-measure tables instead use powers of a single phase; their linear cases give the $n=2$ boundary at orders $2$ and $3$.

Python (mpmath)

# For n = 3, use BBSW Theorem 4.1, equation (23).
# The printed digits are heuristic; raise precision and check agreement
# when increasing the logarithmic moment order k.
import mpmath as mp
mp.mp.dps = 150  # decimal working digits

def W3(s):
    a = 1 + s/2
    return (mp.sqrt(3) * mp.power(3, s+1) * mp.gamma(a)**2
            / (2*mp.pi*mp.gamma(s+2))
            * mp.hyper([a, a, a], [1, (s+3)/2], mp.mpf(1)/4))

k = 2  # vary k to compute another order
print(mp.nstr(mp.diff(W3, 0, k), 100))

David Borwein, Jonathan M. Borwein, Armin Straub and James Wan, Log-sine evaluations of Mahler measures, II, Integers 12 (2012), 1179–1212. (arXiv) (doi)

Jonathan M. Borwein, Armin Straub, James Wan and Wadim Zudilin, with an appendix by Don Zagier, Densities of short uniform random walks, Canadian Journal of Mathematics 64 (2012), 961–990. (arXiv) (doi)

Author preprint of Log-sine evaluations of Mahler measures, II

Densities of short uniform random walks, arXiv version 2

Multiple Mahler measures of $1+x+y_j$ and $1+x+y_j+z_j$ —   Joint logarithmic moments of different forms whose factors share one phase, rather than repeated factors of the same form.

Mahler measures of $1+x_1+\dots+x_{n-1}$ (uniform random walks in the plane) —   The first logarithmic moments of the same independent-phase linear forms.

Moments of uniform random walks in the plane —   Radial power moments $W_n(s)$; the present logarithmic moments are derivatives at $s=0$.

Higher Mahler measure of cyclotomic polynomials —   Higher logarithmic moments of cyclotomic polynomials; the linear case supplies the two-summand boundary.

Higher Mahler measures of $(x^n-1)/(x-1)$ —   Higher logarithmic moments of one-variable geometric sums; the length-two case is the two-summand boundary.

Values of the polylogarithm $\mathrm{Li}_s$ at rational arguments —   The polylogarithm value in the four-term second-moment evaluation.

Values of the Riemann zeta function at rational numbers —   The zeta values in the four-term second-moment evaluation.

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds $n=3,4$ and every $2\leq k\leq8$, covering both short-walk geometries with explicit densities and both parities of the logarithmic moment order. Eight is a noncanonical practical cutoff that extends beyond low-order closed forms while retaining independent checks. The two-summand boundary belongs to the existing one-variable companions, and higher summand counts require additional density analysis. The attached argument-selection.md explains these discrete families and bounds)

How they were obtained:

The three-term values are Taylor derivatives of the hypergeometric generating function in Borwein, Borwein, Straub and Wan. The four-term values integrate convergent local series for the radial density from Borwein, Straub, Wan and Zudilin. Computations at 150 and 220 decimal working digits agree by at least 148 decimal places relative to max(1, absolute value). The entries retain 100 significant digits. Arb controls finite arithmetic, but the series tails are not certified, so the precision claim is heuristic agreement, not a rigorous enclosure of the infinite computation.

more

Independent elliptic-density quadrature checks every three-term value to 115 decimal places. For four terms, the second-moment closed form is checked to 120 places and direct hypergeometric density evaluations check each series region to 115 places. Separate numerical integration checks all four-term moments to 40 places; that integration shares the density recurrence and checks the integration arithmetic. Known normalization, squared-radius and first-moment controls are run first. The attached method.md explains the algorithms, checks, limitations and a resolved sign discrepancy in a printed endpoint expansion. The attached checks.py reproduces the validation.