URMS2 attack: the resummed mean-square bridge

Disposition

This attack closes at Outcome D, resummed barrier sharpened.

No positive zero-statistics band is obtained, including at level one. The largest band currently promoted is therefore alpha_0=0.

The failure is narrower than in the historical argument. The exact inverse is bounded in a natural weighted Wiener algebra, acts boundedly on the companion Dirichlet Hilbert space, has a controlled height derivative, and has a convergent reflected coefficient envelope for alpha<1/2 at level one and alpha<1/4 at level two. Montgomery-Vaughan controls the ordinary off-diagonal mean-value error once the coefficients are frozen.

What remains missing is one full logarithm in the resummed diagonal square sum, together with an early-smoothed contour comparison for the full inverse. The elementary estimate is of size

\[ x(\log x)^2, \]

while the required scale is

\[ x\log x. \]

This is already a barrier for URMS1. The higher nonlinear denominator is not yet the first unresolved difference between levels one and two.

1. Three precise targets

For kappa=1,2, let

\[ G_\kappa(s)=\frac{\xi^{(\kappa+1)}(s)} {\xi^{(\kappa)}(s)} \]

and let gamma^(kappa) denote the ordinates of the zeros of xi^(kappa), with multiplicity. Assume RH throughout this section. Fix 5/4<sigma<2, put s=sigma+it, and use the pinned Cauchy kernel

\[ k(w,s)=\frac{2\sigma-1} {(w-(s-1/2))(w-(1/2-\bar s))}. \]

Define

\[ Z_{\kappa,x}(s)= (2\sigma-1)\sum_{\gamma^{(\kappa)}} \frac{x^{i\gamma^{(\kappa)}}} {(\sigma-1/2)^2+(t-\gamma^{(\kappa)})^2}. \]

Let a_kappa(n,s) be the exact height-dependent coefficient family of the untruncated right-half-plane inverse. Its two-range transform is

\ \begin{split} \mathcal D_x[a_\kappa (s)=x^{-1/2}\bigg(& \sum_{n\le x}a_\kappa(n,1-\bar s)(x/n)^{1-\bar s}\\ &+\sum_{n>x}a_\kappa(n,s)(x/n)^s\bigg). \end{split} \]

Let P_kappa,x(s) be the explicit diagonal, archimedean, and crossed-pole terms from the same Cauchy transform. The error to be controlled is

\ E_{\kappa,x}(s)=Z_{\kappa,x}(s)-P_{\kappa,x}(s) -\mathcal D_x[a_\kappa (s). \]

This definition prevents the spike and archimedean main term from being misclassified as an error.

A. Pointwise-band URMS2

For every compact K subset (0,1), with x=T^alpha, prove

\[ \sup_{\alpha\in K}\frac1{T\log T} \int_T^{2T}|E_{2,T^\alpha}(\sigma+it)|^2dt\longrightarrow0. \]

Local uniformity, rather than endpoint uniformity, is the requested mode.

B. Weighted URMS2

Let

\[ A_v(\alpha)=\int_{\mathbb R}v(u)v(u-\alpha)du \]

for the exact rational window in window_certificate.py. The smaller target is

\[ \frac1{T\log T}\int_0^1 A_v(\alpha) \int_T^{2T}|E_{2,T^\alpha}(\sigma+it)|^2dt\,d\alpha \longrightarrow0. \]

The alpha integral is inside the analytic statement. No pointwise contour bound is inserted first.

C. Narrow-band URMS2

For one fixed alpha_0>0, prove

\[ \sup_{0\le\alpha\le\alpha_0}\frac1{T\log T} \int_T^{2T}|E_{2,T^\alpha}(\sigma+it)|^2dt\longrightarrow0, \]

after the explicit P_2,x subtraction at alpha=0. The natural first target from the exact coefficient algebra is any alpha_0<1/4.

The same three statements with subscript one define URMS1. Its natural coefficient-algebra target is alpha_0<1/2.

2. Exact convolution inverse

For sigma>1, define the weighted Wiener norm

\[ \|a\|{1,\sigma}=\sum{n\ge1}|a(n)|n^{-\sigma}. \]

Dirichlet convolution makes this a Banach algebra. Put

\[ M_j(\sigma)=\sum_{n\ge2} \frac{\Lambda(n)(\log n)^j}{n^\sigma}. \]

Every M_j(sigma) is finite. For level two, write

\[ A_0=L^2+L',\qquad A_+=2LD+D^2+D'. \]

Then

\[ \|A_+\|_{1,\sigma} \le 2|L|M_0+M_0^2+M_1. \]

Whenever

\[ \delta_2= \frac{2|L|M_0+M_0^2+M_1}{|L^2+L'|}<1, \]

the full inverse exists in the Wiener algebra and satisfies

\[ \|b\|_{1,\sigma} \le\frac1{|L^2+L'|-2|L|M_0-M_0^2-M_1}. \]

Since |L(s)| grows like one half of log |t|, this condition holds uniformly on the right contour line above an epsilon-dependent height. It is an estimate for the full inverse, not a finite Taylor polynomial.

For

\[ N_+=2LD'+2DL'+2DD'+D'', \]

one has

\[ \|N_+\|_{1,\sigma} \le2|L|M_1+2|L'|M_0+2M_0M_1+M_2. \]

Together with N_0=2LL'+L'', this gives

\[ \|a_2\|_{1,\sigma} \le M_0+(|N_0|+\|N_+\|{1,\sigma})\|b\|{1,\sigma}. \]

Level one is simpler:

\[ \|(L+D)^{-1}\|_{1,\sigma}\le(|L|-M_0)^{-1} \]

once |L|>M_0.

Support and multiplicativity

The recursion

\[ b(n,s)=-A_0(s)^{-1} \sum_{\substack{d\mid n\\d>1}}a_+(d,s)b(n/d,s) \]

is finite at every n. Since D is supported on prime powers and D^2 on products of two prime powers, repeated convolution gives the inverse unbounded almost-prime support.

The normalized inverse is not multiplicative. For the first coprime pair 2,3, the frozen level-two denominator gives the exact defect

\[ b(6)-b(2)b(3) =\ell_2\ell_3z^2 (\ell_2\ell_3z^2-2\ell_2z-2\ell_3z+2), \]

which is not the zero polynomial. Thus Euler-product methods do not apply to the inverse sequence without additional structure.

Height derivative

Differentiating the convolution identity A*b=1 gives

\[ \partial_t b=-b*(\partial_t A)*b \]

and hence

\[ \|\partial_t b\|{1,\sigma} \le\|b\|{1,\sigma}^2 \|\partial_t A\|_{1,\sigma}. \]

For two heights, the resolvent identity similarly gives

\[ \|b_t-b_u\|_{1,\sigma} \le\|b_t\|_{1,\sigma}\|A_t-A_u\|_{1,\sigma} \|b_u\|_{1,\sigma}. \]

These identities isolate height dependence without differentiating a coefficient-order expansion.

3. Natural Hilbert space

The mean-square companion is

\[ \mathcal H^2_\sigma= \left\{a:\|a\|{2,\sigma}^2 =\sum{n\ge1}|a(n)|^2n^{-2\sigma}<\infty\right\}. \]

Young's inequality on the multiplicative semigroup gives

\[ \|a*c\|{2,\sigma} \le\|a\|{1,\sigma}\|c\|_{2,\sigma}. \]

Therefore the exact inverse acts as a bounded multiplier on H^2_sigma, with operator norm at most its Wiener norm. Multiplication by D,D',D'' is controlled by M_0,M_1,M_2, respectively.

This space solves the right-half-plane resolvent problem. It does not by itself evaluate the reflected polynomial in the first range n<=x, where the coefficient weights grow with log n/L. The Hilbert multiplier theorem and the arithmetic square-density theorem are different obligations.

4. The narrow-band base camp

Put

\[ \lambda_T=\frac12\log\frac{T}{2\pi}, \qquad r=\frac{\log n}{\lambda_T}. \]

For an operator word beta, the elementary convolution bound is

\[ |W_\beta(n)| \le(\log n)^{\operatorname{length}(\beta)+|\beta|}. \]

Every word in q_j has length plus logarithmic degree j+1.

At level one, the absolute word mass is one at every order, so the full coefficient envelope is

\[ |a_{1,T}(n)|\le(\log n)G_1(r), \qquad G_1(r)=\frac1{1-r},\quad r<1. \]

For n<=T^alpha, this is available on every compact alpha<1/2.

At level two, the exact word masses are

\[ 1,2,3,6,12,\ldots, \]

and give

\[ |a_{2,T}(n)|\le(\log n)G_2(r), \]

where

\[ G_2(r)=1+2r+\frac{3r^2}{1-2r},\qquad r<\frac12. \]

Thus the exact coefficient sequence is pointwise resummed on every compact alpha<1/4.

This does not complete the mean square. Summing the pointwise bound over all integers gives only

\[ \mathcal A_{\kappa,T}(x) :=\sum_{n\le x}|a_{\kappa,T}(n)|^2 \le x(\log x)^2G_\kappa(r_x)^2. \]

The target scale is x log x. After division by that scale, the available bound is

\[ (\log x)G_\kappa(r_x)^2, \]

which diverges on every fixed positive alpha band. The exact coefficient threshold therefore does not by itself imply narrow-band URMS2.

5. The missing logarithm

Let F_kappa(alpha) denote the regular coefficient-generated form-factor object and define its square-density function by

\[ \Phi_\kappa(r)=\frac{2}{r}F_\kappa(r/2), \]

with the value at zero given by continuity. The arithmetic estimate needed by the two-range Stieltjes calculation is

RAMS-kappa(alpha_0)

For x=T^alpha, uniformly on 0<=alpha<=alpha_0,

\[ \mathcal A_{\kappa,T}(x) =x\log x\,\Phi_\kappa(\log x/\lambda_T) +o(x\log x). \]

A directly weighted version, sufficient for the mean square, is

\[ x^{-2}\int_{1-}^{x}u\,d\mathcal A_{\kappa,T}(u) +x^2\int_x^\infty u^{-3}\,d\mathcal A_{\kappa,T}(u) =\log x\,\Psi_\kappa(\alpha)+o(\log T), \]

where Psi_kappa is the exact Stieltjes transform of Phi_kappa.

The second statement is smaller because it asks only for the quadratic form used downstream. Either statement must save exactly one logarithm over the current elementary estimate.

Farmer-Gonek-Lee establish the corresponding coefficient pairings at every fixed order. Their estimates do not state constants uniform in a growing order, so they do not imply RAMS1 for the full inverse. The corrected level-two coefficient derivation has the same limitation.

6. A slowly growing projection is sufficient but not yet available

Fix alpha_0<1/4 and put

\[ \rho=4\alpha_0<1. \]

The elementary square tail beyond operator order J has normalized scale

\[ O((\log T)\rho^{2(J+1)}). \]

It is enough to take

\[ J(T)>\frac{\log\log T}{|\log\rho|}+O(1). \]

This does not redefine the full inverse as a finite expansion. It is a Hilbert-space projection used only to estimate its tail.

For example, when rho=4/5 and the numerical value of log T is 10^6, the smallest integer selected by the exact inequality

\[ (\log T)\rho^{2(J+1)}\le(\log T)^{-1} \]

is J=61.

Closing the argument this way requires the almost-prime square-mean errors to remain summable uniformly through J(T). The pinned source supplies only fixed-order constants. Sathe-Selberg type counting controls the number of almost primes in a growing-order range, but no located theorem supplies the log-weighted, correlated operator-word square asymptotic RAMS1 or RAMS2.

7. Mean-value and off-diagonal terms

Once the height-dependent coefficients are replaced by a frozen sequence, the Montgomery-Vaughan mean-value theorem gives, schematically,

\[ \int_T^{2T}\left|\sum_{n\le x}c_n n^{-it}\right|^2dt =T\sum_{n\le x}|c_n|^2 +O\left(x\sum_{n\le x}|c_n|^2\right). \]

For x=T^alpha, the error is smaller than the diagonal by T^(alpha-1) throughout every compact alpha<1. Applying this theorem to the full frozen coefficient sequence avoids the historical factor (log x)^(4B).

Thus the ordinary Dirichlet-polynomial off-diagonal is not the narrow-band barrier. The missing input is the arithmetic size and asymptotic shape of sum |c_n|^2 itself.

8. Archimedean freezing

Across t in [T,2T],

\[ L(\sigma+it)-\lambda_T=O(1), \qquad L'=O(T^{-1}),\qquad L''=O(T^{-2}). \]

Inside the strict coefficient radii, differentiating the resummed envelopes gives

\[ |a_\kappa(n,L(t))-a_\kappa(n,\lambda_T)| \le C_{\kappa,\alpha_0}\frac{\log n}{\lambda_T}. \]

The square of this difference sums to O(x) by the elementary pointwise bound. If RAMS-kappa supplies sum |a_kappa|^2=O(x log x), Cauchy-Schwarz makes the cross term o(x log x). The L' and L'' terms are smaller.

Therefore freezing introduces no additional power loss once the missing one-log square-mean bound is available. Without RAMS-kappa, the current pointwise estimate cannot make the cross term negligible. Freezing is not silently counted as closed before that input exists.

9. Long Dirichlet tail

On Re(s)=1+epsilon, the exact inverse belongs to the Wiener algebra. Hence

\[ \sum_{n>X}|a_\kappa(n,s)|n^{-1-\varepsilon}\longrightarrow0 \]

uniformly in height after the right-line inverse gap is fixed. The same holds in H^2_(1+epsilon).

This controls a tail beyond a freely chosen X, not the entire second range n>x, which contributes to the main Stieltjes quadratic form. A valid proof must first retain that range and then let X/x grow. The Wiener result prevents an uncontrolled infinite coefficient tail, but it does not evaluate the main two-range mean square.

10. URMS1 calibration

Level one has all the structural advantages expected:

• inverse denominator L+D rather than U^2+U';
• coefficient envelope radius alpha<1/2;
• word mass one at every order;
• the same Wiener and Hilbert multiplier bounds;
• the same Montgomery-Vaughan reduction after freezing.

Nevertheless, the elementary square sum is still only

\[ x(\log x)^2G_1(2\alpha)^2, \]

and the required RAMS1 scale is x log x. No independent uniform almost-prime square asymptotic was located. URMS1 is therefore not obtained, and Outcome B is not claimed.

This identifies the present ordering of difficulties:

1. uniform resummed almost-prime square density, already at level one;
2. exact early-smoothed contour transfer;
3. the additional level-two nonlinear numerator and denominator.

11. Exact contour target

The old false estimate is not repaired or reused. The replacement contour target is

RC-kappa(alpha_0)

For the exact full inverse and x=T^alpha,

\[ \sup_{0\le\alpha\le\alpha_0} \frac1{T\log T}\int_T^{2T} |E_{\kappa,T^\alpha}(\sigma+it)|^2dt=o(1). \]

The high right contour line is controlled by the Wiener inverse. What remains is an early-smoothed passage through the Cauchy transform that retains cancellation, handles the reflected finite range, and sums the residues of the coefficient functions without a pointwise absolute tail estimate.

The permanent planted failure remains

\[ \int_V^\infty\frac{dv}{1+(v-t)^2}\ge\frac12 \quad(t\ge V). \]

Nothing in the Wiener or Hilbert argument implies decay for this integral. The new architecture therefore does not reproduce the known false step.

12. Smallest sufficient package

For any fixed alpha_0<1/4, the following two estimates would give narrow-band URMS2:

1. RC-2(alpha_0), the exact early-smoothed contour comparison;
2. the weighted RAMS2 Stieltjes asymptotic with error o(log T).

The existing exact inverse, Hilbert multiplier, long-tail, projection, and freezing bounds then control every remaining passage. For URMS1, replace 1/4 by 1/2 and use the level-one versions of the same two estimates.

Current estimates miss RAMS-kappa by exactly the exponent change

\[ x(\log x)^2\quad\hbox{to}\quad x\log x. \]

They miss RC-kappa qualitatively: right-line norm control is available, but no published result located here turns it into the required reflected, early-smoothed L2 comparison.

13. Falsification controls

urms2_attack.py records:

• exact finite convolution inversion;
• the smallest multiplicativity defect at 2,3;
• Wiener inverse, resolvent-difference, and height-derivative bounds;
• the level-one and level-two coefficient envelopes;
• the one-log square-loss calculation;
• the J(T) projection scale;
• an independent finite comparison between the exact level-two recurrence and the corrected operator-word projections.

In the default finite comparison, the maximum coefficient error falls monotonically from about 1.70 at order zero to below 1e-12 at order ten. This checks the resummation normalization. It is not an asymptotic input.

The level-one reduction and the old contour-kernel counterexample remain in test_higher_xi.py.

14. Literature boundary

No located source supplies RC1, RAMS1, RC2, or RAMS2 as stated.

15. Trust boundary

Exact algebra and deterministic computation cover the convolution recurrence, nonmultiplicativity witness, norm identities, envelope sums, projection scale, and finite resummation comparison.

Classical analysis is still required for RC-kappa and RAMS-kappa. Numerical CUE and finite-height recurrence data do not enter the disposition.

No simplicity percentage, zeta transfer, or higher derivative is opened.

Pinned state

• Repository state at the start of this attack: 64679e742dd74af9589212ba176c2604fd1c1b13.
• Farmer-Gonek-Lee source: arXiv:0803.0425 (https://arxiv.org/abs/0803.0425).
• Bian thesis PDF SHA-256: ec1143f4f6c83288b717cfd4cd0aa6cc620f8c68b892f510a2a8d1708f36bfb7.

Reproduction

.venv/bin/python hunts/higher_xi/urms2_attack.py
.venv/bin/python -m pytest -q -n0 hunts/higher_xi/test_higher_xi.py