This is a derivation check on a polylog-sized range of denominators, not a proof attempt and not a claim about RH. It does not close rank 3 of the doors table (RESULTS.md, "The doors"; priced in RANK3_SCOPE.md), because the range it reaches, \(2\le q\le\lfloor L^B\rfloor\) for a fixed \(B\), is a bounded power of \(\log N\), while rank 3 runs up to \(R_0\sim N^{1/2}/(3L)\), a positive power of \(N\). RANK3_SCOPE.md Section 2 already states this gap in one line ("(17) covers only a vanishing initial segment of \(2\le q\le R_0\), and Section 6 does not attempt even that"); this document is exactly that attempt, worked out on the segment RANK3_SCOPE.md left untried, and it finds a split answer: yes for the mixed moment, no for the fourth moment, and the reason for the split is structural, not a matter of pushing the same computation harder.
1. Which two objects, at which radius
UPPER_BOUND.md Section 3, equation (7), defines the arcs \(I_{q,a}\) and their radius \(\delta_q=Q/(qN)\) in terms of a parameter \(Q\) chosen once per section. Section 5 sets \(Q=\lfloor L^B\rfloor\) for a fixed \(B\), proves (17) uniformly for \(q\le Q\) on exactly these arcs, and closes (19). Section 6 then resets \(Q=\lfloor\sqrt N/3\rfloor\) and defines \(U_Q\), \(Z_Q\) (just before (23)) as sums over the arcs \(I_{q,a}\) at this wider \(Q\). The same letter \(Q\) names two different radii; the task at hand is to check whether the narrow-arc estimate (17)-(19) can be spliced onto the wide-arc objects \(U_Q\), \(Z_Q\), restricted to \(q\le L^B\), and following the document's own notation convention for restricting these sums by denominator range (\(U_{q>R_0}\) in (26), \(Z_{q>R_0}\) in (28)), write \[ U_{2\le q\le L^B}=\sum_{2\le q\le L^B}\sum_a^\int_{I_{q,a}}|P_{q,a}|^2|R_{q,a}|^2, \qquad Z_{2\le q\le L^B}=\sum_{2\le q\le L^B}\sum_a^\int_{I_{q,a}}|R_{q,a}|^4, \] with \(I_{q,a}\) the wide (square-root) arc, radius \(\sqrt N/(3qN)\).
This choice of reading matters, and the alternative is worth ruling out explicitly. If instead \(U_{2\le q\le L^B}\), \(Z_{2\le q\le L^B}\) meant the same sums over the narrow arcs (radius \(L^B/(qN)\), Section 5's own arcs), the question would already be answered: that is exactly the computation Section 5 performs to reach (19), since (18) integrated over those arcs with the pointwise bound from (17) is the proof of (19), and splitting it into a \(P^2R^2\) piece and an \(R^4\) piece changes nothing about the two terms already displayed there (Section 3 below reproduces this). Reading the question that way would make it a restatement of work already in the text. The wide-arc reading is the one that says something new, because \(U_Q\), \(Z_Q\) at \(Q=\lfloor\sqrt N/3\rfloor\) are the objects Section 6 and (23) actually use, and (17) was never proved on arcs that wide. The rest of this document uses the wide-arc reading.
For \(2\le q\le L^B\), since \(L^B=o(\sqrt N)\) for every fixed \(B\), the narrow arc sits strictly inside the wide arc at the same center: writing \(J_{q,a}=\{\beta:|\beta|\le L^B/(qN)\}\) for the piece (17) controls, \(J_{q,a}\subset I_{q,a}\) for all large \(N\). Split each wide-arc integral into the core \(J_{q,a}\) and the leftover annulus \(I_{q,a}\setminus J_{q,a}\); this is an exact decomposition of the same integrand over a disjoint union of domains, not an approximation. The rest of this note bounds each piece.
2. The core: (17)-(18) reproduce (19)'s two terms, separately, for free
On \(J_{q,a}\), (17) gives \(R_{q,a}(\beta)=F_N(a/q+\beta)-P_{q,a}(\beta) =O_{B,H}(NL^{-H})\), uniformly in \(q\le L^B\), \(a\), and \(\beta\in J_{q,a}\), for every fixed \(H\). This bounds \(P_{q,a}^2R_{q,a}^2\) and \(R_{q,a}^4\) directly, without needing (18) at all — (18) is only needed once \(R_{q,a}\) is not already small, which is the annulus's problem, handled in Section 3: \[ \int_{J_{q,a}}|P_{q,a}|^2|R_{q,a}|^2\ll_{B,H}N^2L^{-2H}\int_{I_{q,a}}|P_{q,a}|^2, \qquad \int_{J_{q,a}}|R_{q,a}|^4\ll_{B,H}N^4L^{-4H}|J_{q,a}|. \] Summing over \(2\le q\le L^B\), \(a\), using \(\sum_{q,a}\int_{I_{q,a}}|P_{q,a}|^2 \ll N\log(2L^B)\) (the estimate right before (19), restricted to \(q\ge2\)) and \(\sum_{q,a}|J_{q,a}|\ll(L^B)^2/N\) (the arc-disjointness bound of Section 3, applied at the narrow radius) gives \[ U^{\rm core}{2\le q\le L^B}\ll{B,H}N^3L^{-2H}\log(2L^B),\qquad Z^{\rm core}{2\le q\le L^B}\ll{B,H}N^3L^{2B}L^{-4H}. \] These are exactly the two terms of (19) (with \(Q=L^B\)), separated rather than added — unsurprising, since (19) is proved by exactly this pointwise bound on \(R_{q,a}\), and \(8U_Q+2Z_Q\ge M_Q\) is how (18) turns a bound on \(R\) into a bound on \(M_Q\) in the first place. So on the core alone, nothing is lost or gained by separating: the two pieces of (19) already are separate bounds for (narrow-arc) \(U\) and \(Z\). The open question is only the leftover annulus.
3. The annulus for \(U\): survives, using \(P\)'s own decay
On the annulus, (17) does not apply, so \(R_{q,a}=F_N-P_{q,a}\) must be bounded via (18)'s style split \(|R_{q,a}|^2\le2|F_N|^2+2|P_{q,a}|^2\) (the same elementary step used before (25)), giving \[ U^{\rm ann}{2\le q\le L^B}\le2\sum{q,a}\int_{\rm ann}|P_{q,a}|^2|F_N|^2 +2\sum_{q,a}\int_{\rm ann}|P_{q,a}|^4 . \] Two ingredients are available, both already in the text. First, Vaughan's (V) applies on the whole wide arc \(I_{q,a}\) for \(q\le L^B\) (its hypothesis \(|\alpha-a/q|\le q^{-2}\) holds since \(\sqrt N/(3qN)\le q^{-2}\) once \(q\le3\sqrt N\), true here), and for \(q\) as small as a fixed power of \(L\), its dominant term is \(Nq^{-1/2}\) (the other two terms are smaller by a positive power of \(N\) once \(q\ll N^{2/5}\), which \(L^B\) certainly is), so \(\sup_{I_{q,a}}|F_N|\ll Nq^{-1/2}L^{5/2}\), depending on \(q\). Second, the elementary kernel bound \(|K_N(\beta)|\le\min(N,(2\|\beta\|)^{-1})\) already used for \(H_Q\) in Section 4 gives, by direct integration from the inner radius \(L^B/(qN)\) outward, \[ \int_{\rm ann}|K_N|^2\ll qN/L^B,\qquad\int_{\rm ann}|K_N|^4\ll(qN/L^B)^3, \] hence \(\int_{\rm ann}|P_{q,a}|^2\ll qN/(\phi(q)^2L^B)\) and \(\int_{\rm ann}|P_{q,a}|^4\ll q^3N^3/(\phi(q)^4L^{3B})\), term by term in \(q\), not merely on average.
Combining the \(q\)-dependent sup from (V) with the \(q\)-dependent kernel tail, term by term: \[ \int_{\rm ann}|P_{q,a}|^2|F_N|^2\ll(Nq^{-1/2}L^{5/2})^2\cdot \frac{qN}{\phi(q)^2L^B}=\frac{N^3L^{5-B}}{\phi(q)^2}. \] Summing over \(a\) (\(\phi(q)\) terms) and then \(q\le L^B\), using \(\sum_{q\le x}1/\phi(q)\ll\log x\), gives \(\sum_{q,a}\int_{\rm ann}|P|^2|F_N|^2 \ll N^3L^{5-B}\log L\). The \(P^4\) term is smaller (it inherits a full \(L^{-3B}\) from the kernel tail against a bounded-average-order sum \(\sum_q(q/\phi(q))^3\ll L^B\), net \(\ll N^3L^{-2B}\)). So \[ U^{\rm ann}_{2\le q\le L^B}\ll_B N^3L^{5-B}\log L . \] This shrinks as \(B\) grows, because both ingredients that produced it — Vaughan's \(q^{-1/2}\) decay and the kernel's tail decay from the inner radius \(L^B/(qN)\) — respond to how far out the good core reaches. Combined with the core bound of Section 2, for every target saving \(H'>0\), first fix \(B\) large enough that \(N^3L^{5-B}\log L\ll N^3L^{-2H'}\), then fix any \(H\ge H'\) so the core term is at least as small: this gives \[ U_{2\le q\le L^B}\ll_{H'}N^3L^{-2H'+o(1)}\quad\text{for every fixed }H'>0, \] matching, up to the \(\log L\) bookkeeping, the strength (19) reaches for the combined \(M_Q\) at \(Q=L^B\). The mixed moment survives the splice.
4. The annulus for \(Z\): does not survive, and cannot be pushed through by raising \(B\)
The same attempt for \(Z\) fails, and fails for a reason that does not go away as \(B\to\infty\). Write \(|R_{q,a}|^4\le8(|F_N|^4+|P_{q,a}|^4)\) (the step used before (28)). The \(P^4\) piece is the same small term as above, \(\ll_B N^3L^{-2B}\)-type. The \(F_N^4\) piece is the obstruction: \[ \sum_{q,a}\int_{\rm ann}|F_N|^4\le\Big(\sup_{\mathbb T}|F_N|\Big)^2 \sum_{q,a}\int_{\rm ann}|F_N|^2\le\Big(\sup_{\mathbb T}|F_N|\Big)^2\int_{\mathbb T}|F_N|^2, \] using that the annuli for distinct \((q,a)\) are disjoint (Section 3) to replace their union by the whole circle. Both factors on the right are already in the text: \(\sup_{\mathbb T}|F_N|\le\psi(N)\ll N\) trivially (Chebyshev, valid everywhere, and for these small, fixed \(q\) this is at least as good as (V), whose first term \(Nq^{-1/2}L^{5/2}\) carries an extra \(L^{5/2}\) that the trivial bound does not), and \(\int_{\mathbb T}|F_N|^2=d_N\ll NL\) (used already before (20)). So \[ \sum_{q,a}\int_{\rm ann}|F_N|^4\ll N^2\cdot NL=N^3L. \] This bound involves neither \(B\) nor \(H\): the trivial sup and the total \(L^2\) mass \(d_N\) are both fixed quantities that do not know where the SW-good core ends. Raising \(B\) pushes the core further out and shrinks the width of the annulus, but neither ingredient used to bound the annulus's \(F_N^4\) content responds to that width — the sup bound is pointwise and the \(d_N\) bound is a bulk total over the entire circle, not something that shrinks as the excluded core grows. So \[ Z^{\rm ann}_{2\le q\le L^B}\ll N^3L,\qquad\text{for every fixed }B, \] and this is the governing term: it is not \(o(N^3L^{-2H})\) for any \(H>0\), so it does not match (19)'s strength, and no larger choice of \(B\) repairs it. The fourth moment does not survive the splice.
5. Why the two moments split apart
The mixed moment has one leg, \(P_{q,a}\), that is a fully explicit function (a normalized Dirichlet kernel), so its mass on any sub-region of an arc — in particular the annulus outside the SW-good core — is computable directly from \(|K_N(\beta)|\le\min(N,(2\|\beta\|)^{-1})\), the same elementary estimate Section 4 already uses for \(H_Q\), and that mass decays as a negative power of the core's own radius \(L^B/(qN)\): raising \(B\) shrinks it as fast as wanted. The fourth moment's leading term has no such leg: both factors of \(|F_N|^4\) are the unknown arithmetic object itself, and the only tools on hand for it outside the SW-good zone are a pointwise bound (Vaughan's (V), which Section 6 and RANK3_SCOPE.md Section 2 both note is no better than trivial at fixed small \(q\)) and a bulk \(L^2\)-mass identity (\(d_N\)) that is not localized to any particular range of \(q\). Multiplying the best pointwise bound by the best bulk bound gives a number that cannot see \(B\) at all. This is the same asymmetry Section 7 already exhibits at \(q=1\): \(U_1\) has an exact convolution/Parseval identity (29) turning it into \(\sum\Delta(t)^2\), which (SW) then bounds by \(O_H(N^3L^{-2H})\); no analogous identity or bound is exhibited anywhere in UPPER_BOUND.md or RANK3_SCOPE.md for a \(Z\)-side object at \(q=1\) or at any \(q\ge2\). The computation above is the same gap, one layer further out.
6. What this does and does not settle
The splice of (17)-(19) onto Section 6's wide-arc moments, restricted to \(2\le q\le L^B\) for a fixed \(B\), gives a separate bound for \(U\) matching (19)'s strength (choosing \(B\) as large as the target saving requires), and fails to give any useful separate bound for \(Z\) — the best bound reached here for \(Z\), \(O(N^3L)\), exceeds \(N^3\), let alone the \(O_\epsilon(N^{2+\epsilon})\) that (23) would need. Both conclusions rest only on tools already present in UPPER_BOUND.md Sections 3-6: (17), (18), (V), the kernel estimate before (13), and Parseval; nothing here assumes a stronger unconditional input than the source text already uses.
This does not close rank 3. Even the successful half, for \(U\), reaches only \(2\le q\le L^B\), a fixed power of \(\log N\), against rank 3's full range up to \(R_0\sim N^{1/2}/(3L)\) — a vanishing fraction of it, in the sense RANK3_SCOPE.md Section 2 already names. It also does not move the total bound (23), independent of that range gap: RANK3_SCOPE.md Section 4 shows rank 1's own reached strength, \(O_H(N^3L^{-2H})\) for the \(q=1\) term \(U_1\), already caps what (23) can reach, and the \(U\)-bound reached here is of that same order, so closing it changes nothing about the sum. Nothing here bears on RH.