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# T706: normalization, formulas, and error bounds
The value is U_N = ||chi+F_N||^2 in L^2((0,infinity),dx), where
chi=1_(0,1), rho_n(x)={1/(nx)}, F_N=sum_(n=1)^N mu(n)rho_n, mu(1)=1,
and N is a positive integer. This is equation (1.1) of Báez-Duarte,
math/0202141v2; equation (1.18) of math/0011254v1 calls the same sum S_n.
Endpoint choices affect no integrals. There is no anchoring, weighted measure,
coefficient optimization, damping, or assumed RH. Logs are real natural logs.
Changing variables t=1/(jx) gives G_jk=A(j/k)/j, with
A(r)=integral_0^infinity {t}{rt}dt/t^2. Also
b_n=(1/n)integral_(1/n)^infinity {t}dt/t^2=(log n+1-gamma)/n.
Indeed the part below 1 is log n, and integration over successive intervals
[m,m+1] telescopes to log(L+1)-H_(L+1)+1 -> 1-gamma. Thus
U_N=1+2 sum mu(j)b_j+sum mu(j)mu(k)G_jk.
For coprime positive p,q, BBLS math/0306251v1, Proposition 89 states
A(p/q)=((1-p/q)log(p/q)+(1+p/q)C)/2
-pi(V(p,q)+V(q,p))/(2q), C=log(2pi)-gamma.
Its §4 defines V(p,q)=sum_(m=1)^(q-1) ((mp mod q)/q)cot(pi m/q),
including V(p,1)=0. Proposition 87 gives A(1)=C. These passages were inspected.
The generator reuses gram_system from T641's generate.py (hash in its docstring)
after checking this formula. All rational arguments are constructed exactly,
then enclosed in Arb; poles are avoided since 0<m<q. No external decimals
are inputs. The incremental quadratic form adds the new row and column at
each N. Its ball encloses the entire finite expression, including cancellation.
Immediate checks: U_1=3+log(2pi)-3gamma. For N=2, G_12=(3C-log 2)/4,
G_22=C/2, hence U_2=2-gamma-(log 2)/2. If mu(N)=0 then F_N=F_(N-1).
For x>1, F_N(x)=g_N/x, g_N=sum mu(n)/n, giving the exact exterior
contribution g_N^2. Choosing c_n=-mu(n) in the minimum defining T641 gives
U_N>=d_N^2. These are algebraic or integral deductions, not conjectures
inferred from finite agreement. No monotonicity or limiting decay is claimed.
## Separate integral calculation
Split A(p/q) into periods t=q(n+u) to obtain
A(p/q)=(1/q)integral_0^1 f(u) sum_(n=0)^infinity(n+u)^(-2)du,
f(u)={qu}{pu}. On the intervals split at i/q and j/p, f is an exact quadratic
alpha*u^2+beta*u+delta. For n>=0 the integral is evaluated using
alpha*(n+u)+(beta-2alpha*n)log(n+u)
-(alpha*n^2-beta*n+delta)/(n+u).
At n=u=0 the polynomial has beta=delta=0, and the integrand is alpha.
For n>=K=16 use the uniformly convergent alternating expansion
(n+u)^(-2)=sum_(m=0)^M (-1)^m(m+1)u^m/n^(m+2)+remainder,
with M=125. Moments of f are exact rationals; the summed coefficients are
Hurwitz zeta(m+2,K) enclosed by Arb. Because 0<=f<=1 and terms decrease,
the omitted absolute integral is bounded by
(M+2)/K^(M+3)+1/K^(M+2), before division by q. This follows from the
first omitted term and the integral bound on the positive zeta tail.
The bound is added to the ball; it is not precision agreement. This kernel
implementation is reused from T641 checks.py, hash recorded in integral_kernel.py.
Reusing code does not make the prior table evidence: every required kernel
is freshly recomputed and compared here, and the derivation is checked here.
The generator uses finite cotangent sums; this route uses piecewise integrals
and a bounded series. Both use Arb elementary functions and the same definitions.
checks.py checks all 465 symmetric Gram entries, assembles every norm via a
full double sum with independently trial-factored Möbius values, and recomputes
all 30 optimal lower bounds by Gaussian elimination with positive pivots.
Known answers for nonzero elementary integrals and A(1), plus deliberately
wrong, wide and NaN inputs, check the comparison machinery before use.
For N=1,2,3,6 an additional whole-residual check sets t=1/x and
L=lcm(1,...,N). The fractional sum f(t)=sum mu(n){t/n} is L-periodic.
Integrate (1_(t>1)+f(t))^2/t^2 directly on [0,L]: [0,1] contributes g_N^2,
and the rest is split at integer jumps. On all subsequent periods use
h(u)=(1+f(Lu))^2 and sum_(k=1)^infinity(k+u)^(-2)/L. Integrate the first
15 terms and use the same remainder proof, multiplied by
(1+sum|mu(n)|)^2/L. This avoids the generating b_n formula altogether.
Exact rational endpoints of the resulting balls are preserved in
numerical-evidence.json. verify_stored.py requires both enclosures to lie
strictly inside every stored decimal's centre +/- one last-place interval;
there is no midpoint-only comparison. It also checks the two closed forms,
repeated endpoints, positivity, and exact exterior contributions. Generator
verify(sample=None) is a separate reproduction check. The independent critic
must still assess mathematics and argument selection; this is build evidence.
Instruction archive: skill-manifest.json, bundle SHA-256
d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.