RAMS1 attack: where the extra logarithm comes from

Status

The source of the elementary x(log x)^2 bound is exact. It is a support overcount, not genuine mass in the level-one resummed coefficient family.

The powerful-squarefree factorization added after this attack closes the remaining mixed-prime uniformity. URMS1-CLOSURE.md gives the resulting full RAMS1 theorem, the early-smoothed RC1 argument, and a rebuilt level-one bridge on every compact |alpha|<1/4. This file remains the derivation ledger for the support and Borel-Hadamard discoveries.

1. The exact level-one object

Let

\[ \Lambda_j=\underbrace{\Lambda*\cdots*\Lambda}_{j\text{ factors}}, \qquad \Lambda_0=\delta_1, \]

and define

\[ \alpha_0=-\Lambda, \qquad \alpha_k=\Lambda_{k-1}*(\Lambda\log),\quad k\geq1. \]

For a frozen positive parameter z=1/L, the full coefficient is

\[ a_z(n)=-\Lambda(n)+\sum_{k\geq1}z^k\alpha_k(n). \]

This is untruncated at the level of the Dirichlet object, but it is a finite sum at each integer:

\[ \alpha_k(n)=0\qquad(k>\Omega(n)). \]

The numerical decomposition in rams1.py therefore includes every nonzero depth. It does not select a geometric order.

At the Dirichlet-series level, if

\[ A(s)=\sum_n\Lambda(n)n^{-s}, \qquad B(s)=\sum_n\Lambda(n)\log(n)n^{-s}, \]

then the same coefficient family is generated by

\[ -A(s)+\frac{zB(s)}{1-zA(s)}. \]

This is the level-one resummed resolvent used throughout the previous audit.

There is a further exact collapse. Symmetry among the k convolution factors gives

\[ \boxed{\alpha_k(n)=\frac{\log n}{k}\Lambda_k(n).} \]

Mark one factor with its logarithm and sum over all possible marked positions. This gives log(n) times the unmarked convolution, and every position has the same contribution. alpha_symmetry_defects() checks the identity as a formal polynomial for all integers through 45 and every possible depth.

2. Exact squarefree identity

Let n=p_1...p_r be squarefree. An ordered convolution word contributing to alpha_k(n) has k-1 factors from Lambda and one factor from Lambda log. Since every factor must be a prime power and n has no repeated prime, such a word exists only when k=r. Summing over the choice of the last prime gives

\[ \boxed{ \alpha_k(n)= \begin{cases} (r-1)!\,(\log n)\displaystyle\prod_{p\mid n}\log p,&k=r,\\ 0,&k\ne r. \end{cases}} \]

Thus different convolution depths never cross on squarefree support. Cross-depth terms are collision terms: they require a repeated prime or a prime power. squarefree_identity_defects() checks this identity as a formal polynomial in independent symbols log(p) for every squarefree integer through 70 and every possible depth.

The prime case is the first and most important instance:

\[ a_z(p)=-\log p+z(\log p)^2. \]

It produces the leading factor

\[ 1-2r+r^2=(1-r)^2, \qquad r=z\log x, \]

after the prime number theorem is applied to the three prime sums.

Pure prime powers also have a closed form. The convolution identity

\[ \alpha_k(p^a)=\binom ak(\log p)^{k+1},\qquad1\leq k\leq a, \]

gives

\[ \boxed{a_z(p^a)=\log p\left((1+z\log p)^a-2\right).} \]

When p^a<=x and r=z log x is bounded,

\[ (1+z\log p)^a\leq e^{az\log p}\leq e^r. \]

Consequently the total square contribution from a>=2 is

\[ O_r\left(\sum_{p^a\leq x,\,a\geq2}(\log p)^2\right) =O_r(x^{1/2}(\log x)^2), \]

which is o(x log x). The exact layer and resummed identities are checked for several prime powers as formal polynomials.

3. The missing logarithm is support density

The elementary majorant uses

\[ |\alpha_k(n)|\leq(\log n)^{k+1} \]

for every integer and then counts all x integers. It already loses one logarithm at depth zero:

\[ \sum_{n\leq x}(\log n)^2\asymp x(\log x)^2, \]

whereas the actual prime-power support gives

\[ \sum_{n\leq x}\Lambda(n)^2=x\log x+O(x). \]

The planted dense-support control in support_loss_control() retains the larger scale. The actual prime-supported control remains near one after division by x log x. This distinguishes the arithmetic mechanism from a toy coefficient family where the second logarithm is real.

The classification requested in the directive is therefore:

There is substantial cancellation in the full coefficient square, but it is structurally understood. The dominant negative cross term is the prime-layer pair alpha_0 alpha_1. For squarefree composites only one positive depth diagonal survives. Cross-depth cancellation beyond the prime layer is confined to repeated-prime collision strata and is lower order at each fixed depth.

4. Auxiliary generating objects for the second moment

4.1 Full Borel-Hadamard object

The symmetry identity gives a multiplicative representation of the full resummation. Define

\[ \mathcal E_\tau(s)=\exp(\tau A(s)) =\sum_n E_\tau(n)n^{-s}. \]

Then

\[ E_\tau(n)=\sum_{k\geq0}\frac{\tau^k}{k!}\Lambda_k(n) \]

is multiplicative. The ordinary convolution inverse is its exact Laplace transform:

\[ \sum_{k\geq0}z^k\Lambda_k(n) =\int_0^\infty e^{-t}E_{zt}(n)\,dt. \]

For n>1, the positive part of the level-one coefficient is

\[ c_z(n):=\sum_{k\geq1}z^k\alpha_k(n) =\log n\int_0^\infty e^{-t}\frac{E_{zt}(n)}t\,dt. \]

The apparent singularity at t=0 is removable because E_0(n)=0 for n>1. This packages every convolution depth before an asymptotic limit is taken.

The local factor is explicit:

\[ \sum_{a\geq0}E_\tau(p^a)X^a =\exp\left(\tau\log p\frac{X}{1-X}\right). \]

Hence the full Hadamard second-moment kernel has the Euler product

\[ J(s;t,u;z) :=\sum_n\frac{E_{zt}(n)E_{zu}(n)}{n^s} =\prod_p\left( \sum_{a\geq0}E_{zt}(p^a)E_{zu}(p^a)p^{-as} \right). \]

Before the explicit -Lambda cross terms are added,

\[ \sum_n\frac{c_z(n)^2}{n^s} =\int_0^\infty\int_0^\infty \frac{e^{-t-u}}{tu}\,\partial_s^2J(s;t,u;z)\,dt\,du. \]

This is the requested generating object for the exact second moment. It includes squarefree integers, repeated primes, prime powers, every convolution depth, and every cross-depth interaction. borel_multiplicativity_defects() checks the multiplicativity of E_tau as a formal polynomial on all coprime pairs with product through 35.

It also explains why a fixed zeta(s)^m G(s) factorization is not the natural target. RAMS1 has the simultaneous scaling

\[ z\asymp(\log x)^{-1},\qquad s-1\asymp(\log x)^{-1}, \]

and the first local prime term contains z^2tu(log p)^2 p^-s. The singularity is a two-parameter scaling kernel, not a fixed power of zeta.

The remaining step is analytic: obtain a uniform Tauberian or Selberg-Delange estimate for this Borel-Hadamard product and justify the t,u integrations at the simultaneous scaling. Applying an absolute Euler-product bound first creates an exponential in tu, which is not integrable against e^(-t-u). That failed majorant is another precise reason the available Wiener bound does not finish RAMS1.

4.2 Squarefree specialization

Introduce independent variables y_p and

\[ P(s,u)=\prod_p\left(1+u(\log p)^2p^{-s}\right), \qquad \Theta=-\frac{d}{ds}. \]

Coefficient extraction gives the exact identity

\[ [u^k]\Theta^2P(s,u) = \sum_{\substack{n\text{ squarefree}\\\omega(n)=k}} \frac{(\log n)^2\prod_{p\mid n}(\log p)^2}{n^s}. \]

Multiplication by (k-1)!^2 z^(2k) gives the squarefree depth-k contribution to the Hadamard square of the resummed coefficient family. This is the squarefree specialization of the full Borel-Hadamard object.

The prime-simplex calculation, or equivalently the fixed-depth arithmetic in Farmer, Gonek, and Lee, gives

\[ \sum_{\substack{n\leq x\text{ squarefree}\\\omega(n)=k}} (\log n)^2\prod_{p\mid n}(\log p)^2 \sim \frac{x(\log x)^{2k+1}}{k!(2k-1)!}. \]

Since

\[ \frac{(k-1)!^2}{k!(2k-1)!} =2\frac{(k-1)!}{(2k)!}, \]

this independently reconstructs the fixed-depth diagonal main term

\[ A_{k,k}(x) \sim2\frac{(k-1)!}{(2k)!}x(\log x)^{2k+1}. \]

The factorial decay of these main coefficients makes their infinite sum convergent for every fixed r=z log x.

There is also an order-uniform upper bound for the entire squarefree part. Chebyshev's estimate supplies a constant C with

\[ \sum_{p\leq e^u}(\log p)^2\leq Ce^u u. \]

Dominating this prime measure by the derivative of Ce^u u, convolving k times in logarithmic coordinates, and dropping the distinctness restriction gives, for another absolute constant D,

\[ \sum_{\substack{n\leq x\text{ squarefree}\\\omega(n)=k}} (\log n)^2\prod_{p\mid n}(\log p)^2 \leq \frac{D^k x(\log x)^{2k+1}}{k!(2k-1)!}. \]

After multiplication by (k-1)!^2 z^(2k), the ratio of successive majorant terms is asymptotic to D r^2/(4k). The series is summable for every bounded r. Thus the all-depth squarefree contribution is rigorously at most O_r(x log x). Together with the closed prime-power estimate above, this leaves only mixed nonsquarefree integers, those with at least two distinct primes and at least one repeated prime, in the RAMS1 uniformity gap.

5. The candidate full square-density function

The fixed-depth main terms assemble to

\[ \Phi_1(r)=1-2r +2\sum_{k\geq1}\frac{(k-1)!}{(2k)!}r^{2k}. \]

For every fixed order K, the source arithmetic gives

\[ \sum_{n\leq x} \left|\sum_{k=0}^Kz^k\alpha_k(n)\right|^2 =x\log x\,\Phi_{1,K}(r)+O_K(x). \]

What is not available is permission to take K through all depths while x grows. The exact finite identity is

\[ \mathcal A_{1,z}(x) =\sum_{k,l\geq0}z^{k+l}A_{k,l}(x), \]

but the published O constants in the fixed-depth estimates are not tracked uniformly in k,l.

6. Measured finite ladders

rams1.py computes the entire finite coefficient family through floor(log_2 x). The following values are raw, with no fitted rescaling:

The ladders are consistent with the x log x candidate and inconsistent with an additional growing logarithm over this range. They are falsification data, not a uniform asymptotic argument.

At r=0.8, the repeated-prime mixed stratum divided by x log x decreases from about 0.00281 at x=1,000 to 0.00155 at x=1,000,000. The higher-prime-power stratum decreases faster. This locates the remaining uniform problem in collision bookkeeping, rather than exposing new leading mass.

7. The exact missing RAMS1 estimate

Write

\[ A_{k,k}(x)= 2\frac{(k-1)!}{(2k)!}x(\log x)^{2k+1}+E_{k,k}(x). \]

A sufficient order-uniform statement on a band 0<=r<=r_0 is

\[ \sum_{k\geq1}r^{2k} \frac{|E_{k,k}(x)|}{x(\log x)^{2k+1}} +2\sum_{k>l\geq0}^{\!*}r^{k+l} \frac{|A_{k,l}(x)|}{x(\log x)^{k+l+1}} \longrightarrow0, \]

uniformly in r. The star removes the explicit (0,0), (1,0), and (1,1) prime terms.

A stronger but easier-to-instantiate schema would supply nonnegative constants M_k and M_k,l such that

\[ |E_{k,k}(x)|\leq M_kx(\log x)^{2k}, \qquad |A_{k,l}(x)|\leq M_{k,l}x(\log x)^{k+l}, \]

and

\[ \sum_kM_kr_0^{2k}<\infty, \qquad \sum_{k>l}M_{k,l}r_0^{k+l}<\infty. \]

The fixed-depth source gives the displayed powers of log x, but not the two summability statements. Its recurrence repeatedly suppresses the dependence of O constants on k,l, and one step invokes an n^epsilon bound with depth fixed. This is the first exact failure point for resummation.

The squarefree generating object supplies factorial decay for the main term. The unresolved work is an equally explicit summable majorant for repeated prime collisions on mixed support, coprimality removals, and unequal-depth collision types. Pure prime powers are already separated by the closed formula in Section 2.

Equivalently, the full generating language gives the following target.

Borel-Hadamard RAMS1 lemma

For some r_0>0, put z=r/log x. Uniformly for 0<=r<=r_0, the Perron or smoothed inverse Mellin transform of

\[ \int_0^\infty\int_0^\infty \frac{e^{-t-u}}{tu}\partial_s^2J(s;t,u;z)\,dt\,du \]

has the x log x main term generated by Phi_1(r), with an o(x log x) remainder. The estimate must be uniform in the Laplace variables strongly enough to justify both integrations. A theorem for each fixed t,u is not sufficient.

8. Interim RC1 reduction, superseded by the closure

Put a=sigma-1/2>0 and

\[ P_a(u)=\frac{2a}{a^2+u^2}. \]

Its Fourier transform is

\[ \widehat P_a(\xi)=2\pi e^{-a|\xi|}. \]

Therefore Plancherel gives the exact contraction

\[ \|P_a*R\|{L^2(\mathbb R)} \leq2\pi\|R\|{L^2(\mathbb R)}. \]

This is the correct replacement for the false pointwise decay assigned to the long Cauchy tail. It preserves the cancellation that the historical absolute bound removed. It also makes the remaining RC1 obligation precise: after the explicit pole and archimedean terms are removed, the exact height-dependent right-line resolvent must be replaced by its two-range frozen Dirichlet object with o(T log T) local square error on an enlarged height interval. Neither the Wiener norm nor the fixed-depth source estimate supplies that local square bound.

At this point in the attack, RC1 was blocked by the same resummed local mean square and by finite-height boundary localization. URMS1-CLOSURE.md resolves both points using the powerful-squarefree RAMS1 bound, the right-line Cauchy partial fraction, and a separate infinite-tail cutoff.

9. Trust boundary and reproduction

Exact finite controls:

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

Full finite ladders:

.venv/bin/python hunts/higher_xi/rams1.py \
  --limits 1000,10000,100000 --ratios 0.25,0.5,0.8

The exact layer recurrence, formal squarefree identity, support partition, depth-matrix reconstruction, and planted dense-support failure are checked by the test suite. The prime-number-theorem asymptotics and the order-uniform summability target are analytic statements. Python is not used to bridge that boundary.

10. Sources

• Farmer, Gonek, and Lee, Pair correlation of the zeros of the derivative of the Riemann xi-function (https://arxiv.org/abs/0803.0425), especially the definitions of alpha_k, Proposition 4.1, and the S_k,l induction.
• Farmer, Gonek, Lee, and Lester, fixed logarithmic-derivative products (https://academic.oup.com/qjmath/article/64/4/1057/1567887), useful for fixed products but not uniform resummation.
• Montgomery and Vaughan, Hilbert's inequality (https://doi.org/10.1112/jlms/s2-8.1.73), used only after the diagonal coefficient square is controlled.

No located source states the order-uniform collision estimate in Section 7.