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# Formulas and independent integration for T711
All logs are natural. Let m_k=M(floor(N/k)), w_k=m_k/k, theta_k=k/N,
for 1<=k<N. Mobius inversion gives
sum_{k=1}^N M(floor(N/k)) = sum_{d=1}^N mu(d) floor(N/d) = 1.
Removing k=N therefore yields sum m_k=0 (also true for N=1).
Thus sum w_k theta_k=0. This is an exact identity, not an inference from
stored examples. The generator independently checks it for each evaluation.
## Generator: Gram norm and Mellin cross term
Put A(r)=integral_0^infinity {t}{rt} dt/t^2. Substitution t=j/(Nx) gives
K_{jk}=(j/N)A(k/j). BBLS arXiv:math/0306251v1 §4 definition of V and §9,
Proposition 89, gives, for coprime p,q>0,
A(p/q)=(1-p/q)log(p/q)/2+(1+p/q)(log(2pi)-gamma)/2
-pi(V(p,q)+V(q,p))/(2q),
V(p,q)=sum_{h=1}^{q-1} ((hp mod q)/q) cot(pi h/q); V(p,1)=0.
All these finite sums are computed in Arb from exact rational arguments.
For 0<theta<=1, substitute t=theta/x into
B(theta)=integral_0^1 {theta/x} log(x) dx.
Using integral_1^infinity {t}t^{-s-1}dt=1/(s-1)-zeta(s)/s,
its expansion at s=1 gives integral {t}/t^2=1-gamma and
integral {t}log(t)/t^2=1-gamma-gamma_1. Hence
B(theta)=theta[gamma+gamma_1-1+(1-gamma)log(theta)-log(theta)^2/2].
The common constant cancels in sum w_k B(theta_k). Consequently
C_N=<R_N,lambda>=1/N sum m_k[(1-gamma)log(k/N)-log(k/N)^2/2],
Q_N=||R_N||^2=sum w_j w_k K_jk,
E_N=2+Q_N-2C_N. The norm of lambda is exactly 2 by elementary integration.
The Mellin identity follows by integrating floor(t) interval by interval for
Re(s)>1 and then continuity at s=1; its derivative is integrable there.
## Checker: direct periodic step integration
Set t=1/(Nx), so dx=dt/(Nt^2). Write F(t)=sum w_k {kt}.
It has period 1; since sum k w_k=0, it is piecewise constant, with jumps
-w_k at j/k, j=1,...,k-1. It is zero on 0<t<1/(N-1).
The target lambda=-log(Nt) on t>1/N. Wherever F is nonzero, the target
therefore has this expression, including each translated period.
For a nonzero constant c on an interval (a,b) in (0,1), put
D=psi(b)-psi(a), and z1(a)=-gamma_1(a), the linear coefficient of
zeta(1+s,a)-1/s. Summing the elementary primitives over translated periods:
Q contribution = c^2 D/N,
-C contribution = c[(log(N)+1)D+z1(b)-z1(a)]/N.
Indeed sum[(m+a)^-1-(m+b)^-1]=D, and
sum[log(m+a)/(m+a)-log(m+b)/(m+b)]=gamma_1(a)-gamma_1(b).
These differences converge absolutely. Thus there is no truncated numerical
tail; the Hurwitz coefficients evaluate the entire endpoint sums.
DLMF 25.11.1 and 25.11.17 give the defining sum and argument derivative;
25.11.3 gives the shift. The generalized Stieltjes expansion follows by
subtracting the simple pole. The implementation evaluates the deflated
zeta series directly, never finite-differences near its pole.
The checker uses an independent divisor-recursion Mobius implementation,
exact rational events, and 500-bit Arb Hurwitz series. It imports none of
the generator. It shares Arb and the definition, so it cannot independently
test the Arb library itself. Every stored interval must contain its narrow
finite checking ball, not merely overlap a floating-point midpoint.
## Controls
N=1,2 have F=0 and E=2. For N=3, F=-1/2 on (1/2,1) and zero otherwise
in one period. The digamma duplication identity gives Q_3=log(2)/6.
The above cancelled Mellin expression gives
E_3=2+log(2)/6-(2log(2)/3)[1-gamma+log(3)-log(2)/2].
The checker tests this nonzero closed form, rejects a value shifted by 0.001,
and rejects NaN and broad balls. See executable check and actual results.
Skill manifest: ../20261110T232855Z-build-skills/manifest.json;
bundle SHA-256 d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.