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# Reproducing the ordinary measure
For independent uniform torus angles u,v,w, put S=2cos(u)+2cos(v).
Jensen's formula for z^2+(S-k)z+1 gives zero when |k-S|<=2, and
acosh(|k-S|/2) otherwise. This handles torus zeros through the ordinary
integrable logarithm, without a principal value or modified measure.
Each summand in S has the arcsine density 1/(pi sqrt(4-s^2)) on (-2,2).
Convolving the two densities gives, on (-4,4),
rho(s) = K(1-s^2/16)/(2*pi^2)
= 1/(4*pi*AGM(1,abs(s)/4)).
Here K(m) uses the parameter m, rather than the modulus. The quartic integral
for the convolution reduces to the complete elliptic integral. Equivalently,
the torus change (u,v)=(a+b,a-b) preserves Haar probability measure and gives
S=4cos(a)cos(b); integrating the product distribution gives the same K formula.
DLMF 19.8.5 gives the AGM equality after converting its modulus to our parameter.
Thus the one-dimensional quadrature in generate.py is an integral against a
probability density, not a new normalization of the Mahler measure.
The integration intervals are [-4,min(4,k-2)] and [max(-4,k+2),4], omitting
empty intervals. Both are split at zero when necessary. The square-root
endpoints use acosh(1+t/2)=2asinh(sqrt(t)/2). The density is evaluated by AGM
to avoid catastrophic rounding of 1-s^2/16 to 1. The integral is computed
with tanh-sinh quadrature at 150 and 210 working decimal digits. The convergence
estimates and agreement provide numerical evidence, not certified error bounds.
## Independent modular computation
Bertin, arXiv:math/0501153, Theorem 1.1, sets
w=(eta(tau)eta(6tau)/(eta(2tau)eta(3tau)))^6,
k=w+1/w, q=exp(2*pi*i*tau),
m(P_k)=Re[-pi*i*tau+
sum_{n>=1} sigma_3(n)/n*(4q^n-8q^(2n)+12q^(3n)-24q^(6n))].
The six CM tau values are taken from Bertin et al., arXiv:1208.6240, section
2.3, with positive imaginary parts. At the other eight arguments, checks.py
solves the eta equation in the branch continuing these cases. It records tau
and the residual of k=w+1/w. Agreement with the ordinary Jensen integral is
checked separately for every argument, including the interior values.
For r=|q| and truncation N, sigma_3(n)<2n^3 gives the tail bound
96*r^(N+1)*[(N+1)^2/(1-r)+2(N+1)r/(1-r)^2+r(1+r)/(1-r)^3].
This bounds the series tail at the numerically computed q. It does not certify
the root-finding error or the whole computation; no proven claim is made.
## Other controls and identities
The density must integrate to 1 and have second moment 4. Those controls are
computed using Carlson's elliptic RF, separately from the generator's AGM.
The k=0 value is compared with 3sqrt(3)L(chi_-3,2)/(4pi), using Hurwitz zeta
functions for the character values 0,1,-1 modulo 3. A deliberately perturbed
control must fail the high-precision comparison.
For every exterior argument k>6, an independent constant-term expansion is
used:
m(P_k)=log(k)-sum_{n>=1} c_n/(2n*k^(2n)),
c_n=binomial(2n,n)*sum_{j=0}^n binomial(n,j)^2*binomial(2j,j).
The coefficients are exact integers, and c_n<=36^n bounds the omitted tail
by r^(N+1)/(2(N+1)(1-r)), r=36/k^2. This expansion is not used at k=6.
The right derivative at 6 is computed by differentiating the Jensen integral.
Putting s=4-u^2 cancels its square-root endpoint singularity. Six times this
derivative is compared with Watson's gamma-product evaluation, normalized
as T393. This is a derivative check, not another quantity stored as a row.
The relation m(P_10)=3m(P_2)+2m(P_0) follows by eliminating L(g_8,3) between
the two evaluations in Bertin et al., section 1. It is checked on computed
values and on values read from the draft. Database decimals are interpreted
as intervals of one last-place unit on each side, not exact decimal numbers.
Changing (x,y,z) to (-x,-y,-z) proves the evenness in k analytically. The
torus sum lies in [-6,6], and all its stationary values are -6,-2,2,6,
by sin(u)=sin(v)=sin(w)=0. These facts establish the parameter convention
and the intrinsic argument group independently of numerical digits.