This is an investigation, not a proof attempt, of one sub-question RANK3_SCOPE.md leaves open. RANK3_SCOPE.md Section 2 shows that none of the three tools UPPER_BOUND.md assembles — Siegel-Walfisz (17), Vaughan's bound (V), the additive large sieve (LS) used dyadically in (24)-(26) — reaches \(U_{(q)}\), \(Z_{(q)}\) at a useful strength anywhere in \(2\le q\le R_0\). The question here is narrower and specific: for \(\lfloor L^B\rfloor<q\le R_0\), any fixed \(B\) (so this excludes both the tiny range (17) already reaches and the range near \(q=1\) that RANK3_SCOPE.md treats as tied to rank 1), is there some other unconditional estimate — a different large-sieve weighting, a different arc-splitting by denominator, or a different tool altogether — that bounds \(U_{(q)}\), \(Z_{(q)}\) summed or maxed over this range?
Two things are brought in below that are not in UPPER_BOUND.md or RESULTS.md: the Bombieri–Vinogradov theorem and the Barban–Davenport– Halberstam theorem (BDH), both classical, unconditional, standard results about primes in arithmetic progressions on average over the modulus. Neither name occurs in either source document (checked directly). Everything else — the definitions of \(U_{(q)}\), \(Z_{(q)}\), \(P_{q,a}\), \(R_{q,a}\), the identity (29)-(30) at \(q=1\), and the estimates (13)-(28) — is taken as already established there and used, not re-derived. This document has no access to outside literature beyond what its author already knows; where that knowledge runs out, it says so rather than guessing a citation.
1. Two readings of the question, and why they are not the same
\(U_Q=\sum_{q\le Q}U_{(q)}\), \(Z_Q=\sum_{q\le Q}Z_{(q)}\) exactly, by disjointness of the arcs (RANK3_SCOPE.md Section 1). What (23) actually needs is the sum \[ S_U(B)=\sum_{\lfloor L^B\rfloor<q\le R_0}U_{(q)},\qquad S_Z(B)=\sum_{\lfloor L^B\rfloor<q\le R_0}Z_{(q)}, \tag{P.1} \] bounded by \(O_\epsilon(N^{2+\epsilon})\); nothing in (23) needs any individual \(U_{(q)}\) or \(Z_{(q)}\) bounded on its own. A bound on \(\max_{\lfloor L^B\rfloor<q\le R_0}U_{(q)}\) would imply a bound on \(S_U(B)\) only after multiplying by the range length \(R_0\), which is far too lossy to be useful; so the max-reading is a strictly harder question than what (23) needs, and the two get different answers below. Both are asked because the assignment names both.
2. The current decomposition already spends a full power of \(N\) on a term neither large sieve nor Vaughan is asked to touch
Both (25) (for \(U\)) and (28) (for \(Z\)) bound \(|R_{q,a}|^2\) or \(|R_{q,a}|^4\) by expanding the triangle inequality: \(|R_{q,a}|^2\le 2|F_N|^2+2|P_{q,a}|^2\), \(|R_{q,a}|^4\le8(|F_N|^4+|P_{q,a}|^4)\). Both expansions produce a term built from \(P_{q,a}\) alone, with no \(F_N\) in it — a term that is not RANK3_SCOPE.md's "tool" question at all, since no estimate for the prime-counting side is needed to bound it: it is a fact about \(K_N\). Here is its size in this range, computed directly from what (13)-(16) already establish, with no import.
Lemma. For every arc \(I_{q,a}\) in the construction (7) (i.e. for every \(q\le Q\)), \[ \int_{I_{q,a}}|K_N(\beta)|^4\,d\beta=\Theta(N^3), \tag{P.2} \] with absolute implied constants.
Proof. Upper bound: \(|K_N(\beta)|\le\min(N,(2\|\beta\|)^{-1})\), and \(\delta_q\ge1/N\) always since \(Q\ge1\), \(q\le Q\). Split the arc at \(|\beta|=1/N\): \[ \int_{I_{q,a}}|K_N|^4\,d\beta \le2\int_0^{1/N}N^4\,d\beta+2\int_{1/N}^{\delta_q}(2\beta)^{-4}\,d\beta =2N^3+\tfrac1{24}(N^3-\delta_q^{-3})\le\tfrac{49}{24}N^3. \] Lower bound: for \(|\beta|\le1/(2N)\), \(|N\beta|\le1/2\) so \(|K_N(\beta)|=|\sin(\pi N\beta)/\sin(\pi\beta)|\ge(2/\pi)N\cdot(1-O(1/N))\) for \(N\) large (the sine-ratio ranges between \(1\) at \(\beta=0\) and \(\sin(\pi/2)/(\pi/2)=2/\pi\) at the endpoint, and \(\sin(\pi\beta)\ge2\beta\) here); hence \(\int_{|\beta|\le1/(2N)}|K_N|^4\,d\beta\ge c\,N^3\) for an absolute \(c>0\) and all large \(N\), and \(1/(2N)\le\delta_q\) always. \(\square\)
Consequently, for every \(q\), summing the self-term over the \(\phi(q)\) reduced residues, \[ \sum_a\int_{I_{q,a}}|P_{q,a}|^4=\frac{\mu(q)^2}{\phi(q)^3}\cdot\Theta(N^3). \tag{P.3} \] (The extra \(\phi(q)^{-1}\) against (P.2)'s \(\phi(q)^{-4}\) comes from summing \(\phi(q)\) equal terms.) Summing over the target range and using (15), \(1/\phi(q)\le\zeta(2)(1+\log q)/q\), so \(1/\phi(q)^3\ll(1+\log q)^3/q^3\): \[ \sum_{\lfloor L^B\rfloor<q\le R_0}\frac{\mu(q)^2}{\phi(q)^3} \ll\sum_{q>L^B}\frac{(1+\log q)^3}{q^3} =O_\epsilon(L^{-2B+\epsilon})\quad\text{for every }\epsilon>0, \tag{P.4} \] (the tail of a convergent series with an \((1+\log q)^3\ll_\epsilon q^\epsilon\) bound absorbed). So the self-term contributes \[ O_\epsilon(N^3L^{-2B+\epsilon}) \tag{P.5} \] to \(S_U(B)\) and (with a larger absolute constant) to \(S_Z(B)\), for every fixed \(B\) and every \(\epsilon>0\). This is not small in the sense (23) needs: it is \(N^3\) with an arbitrary but fixed log-power discount, never a saving of any power of \(N\) — the same shape as rank 1's own reached bound \(O_H(N^3L^{-2H})\) (UPPER_BOUND.md Section 7). This piece of the existing decomposition, alone, already exceeds the \(N^{2+\epsilon}\) target throughout the polynomial range, for every fixed \(B\), before any large sieve or Vaughan estimate is applied to the cross term. No tool named in UPPER_BOUND.md is aimed at this term — (24)-(28) are all applied to the \(F_N\)-side, not the \(P_{q,a}\)-side — and none is needed to see that it fails here: (P.2) is a fact about \(K_N\) alone.
3. Running the cited large-sieve argument itself down to \(q=\lfloor L^B\rfloor\)
Section 2 leaves open whether some other splitting of (24)-(26) could do better on the cross term even though the self-term already fails; the answer does not depend on that, but it is worth recording that the cited tool, run as far as it goes, does not either. RANK3_SCOPE.md Section 2 computes that the single dyadic block \(R=1\) (\(q=2\)) already costs \(O(N^3L^3)\) via (25). The same telescoping argument that proves (26), applied to the blocks \(R=2^j\lfloor L^B\rfloor\), \(j=0,1,2,\dots\), up to \(R_0\) instead of starting at \(R_0\), sums the per-block bound (25), \(U_{R<q\le\min(2R,Q)}\ll w(Q)^2N^3L/R^2\), over this geometric sequence of blocks. Since the terms decay geometrically in \(j\), the sum is dominated by its first term, and using \(w(Q)\ll L\) by (15), \[ \sum_{\lfloor L^B\rfloor<q\le R_0}U_{(q)} \ll w(Q)^2N^3L\sum_{j\ge0}(2^j\lfloor L^B\rfloor)^{-2} \ll w(Q)^2N^3L^{1-2B} \ll N^3L^{3-2B}, \tag{P.6} \] exactly the mechanism RANK3_SCOPE.md's Section 2 identifies for the single block at \(q=2\), run one level further down instead of stopping at \(q=2\). So the cited tool, pushed as far into the polynomial range as it will go, gives the same order as Section 2's finding above: \(O(N^3L^{3-2B})\), a fixed log-power saving off \(N^3\) and no saving of any power of \(N\), for every fixed \(B\). Extending the dyadic descent past \(R_0\) is not blocked by anything specific to \(R_0\) — \(R_0\) was chosen for the \(q>R_0\) regime, not because the argument breaks at that point — but it does not reach the target either, for the identical reason (P.5) does not.
Both (P.5) and (P.6) hold throughout the whole range \(\lfloor L^B\rfloor<q\le R_0\), not only near its lower edge: the dyadic sum in (P.6) and the tail sum in (P.4) are each dominated by their term at \(q\approx L^B\) regardless of where the range's upper endpoint (\(R_0\) or \(Q\)) is placed, since the summands decay geometrically (or as a convergent power series) in \(q\). So neither finding is an artifact of stopping at \(R_0\); raising the upper cutoff would not change either order.
4. A cheap, unconditional fact about the whole fourth moment, and why it does not help \(Z\) either
For \(Z\), one might hope to avoid (28)'s weak point — Vaughan's \(N^{4/5}\) term, which RANK3_SCOPE.md Section 2 shows is at its worst on exactly the lower part of this range — by bounding \(\sum_a\int_{I_{q,a}}|F_N|^4\) directly, the way (24)-(25) bound \(\sum_a\int_{I_{q,a}}|F_N|^2\) via (LS), rather than through Vaughan's pointwise sup. This does not obviously help, for an elementary reason independent of any large sieve or zero estimate. Writing \(r(m)=\sum_{1\le n\le N-1}\Lambda(n)\Lambda(m-n)\ge0\) for \(2\le m\le2N\) (with the convention \(\Lambda(n)=0\) outside \([1,N]\)), Parseval gives \(\int_{\mathbb T}|F_N|^4=\sum_mr(m)^2\). Cauchy-Schwarz on the sum defining \(r(m)\) gives \(r(m)\le d_N\ll NL\) (Chebyshev), and summing over \(m\) first, \(\sum_mr(m)=\big(\sum_n\Lambda(n)\big)^2= \psi(N)^2=O(N^2)\) (Chebyshev again). Hence \[ \int_{\mathbb T}|F_N|^4=\sum_mr(m)^2\le\Big(\max_mr(m)\Big)\sum_mr(m) =O(N^3L). \tag{P.7} \] This bound uses nothing beyond Cauchy-Schwarz and \(d_N\ll NL\), already cited in UPPER_BOUND.md Section 6. It shows the whole-circle fourth moment is itself only \(O(N^3L)\) — the same order as the target the individual arcs would need to beat, not smaller — so bounding \(\sum_a\int_{I_{q,a}}|F_N|^4\) by any fraction of (P.7) gives nothing useful unless that fraction is shown to be a genuinely small (\(N^{-1+ \epsilon}\)-order) share of the total, which is exactly the kind of concentration statement (28)'s Vaughan-sup argument attempts and RANK3_SCOPE.md Section 2 already shows fails on the lower part of this range. A large-sieve treatment of \(F_N\) directly, in place of Vaughan's, would have to supply that concentration statement itself; nothing computed here supplies it, and this document does not find it supplied elsewhere.
5. The exact-identity route (Route D), and what it actually needs
RANK3_SCOPE.md's Route D observes that Section 7's identity (29)-(30) for \(U_1\) plausibly generalizes, per residue class, to a per-\((q,a)\) object built from \(\Delta(t;q,a)=\psi(t;q,a)-t/\phi(q)\), and that turning such an identity into a bound "needs a uniform estimate for \(\Delta(t;q,a)\) at moduli growing like a power of \(N\)" — the same requirement as Route A. This section asks what "uniform" needs to mean, using the one place the construction actually carries this out: \(q=1\).
At \(q=1\), Section 7 states the mechanism exactly: (SW) gives \(\max_{t\le N}|\Delta(t)|\ll_HNL^{-H}\) pointwise in \(t\), and feeding this into the exact identity (29) — a sum over \(N\) values of \(t\) — costs \[ \sum_{t=1}^N\Delta(t)^2\le N\max_{t\le N}\Delta(t)^2\ll_HN^3L^{-2H}, \tag{P.8} \] which is exactly Section 7's stated bound and exactly the reason it is "a power of \(N\) worse" than (31)'s target. This is not a weakness of (SW)'s strength in \(q\); it is what happens to any pointwise-in-\(t\) bound once it is squared and trivially summed over \(N\) values of \(t\). Reaching (31) needs cancellation summed over \(t\), not a uniform pointwise bound propagated through the sum.
This generalizes directly: whatever tool supplies control on \(\Delta(t;q,a)\) — (SW) up to \(q\le L^B\), or, hypothetically, some tool reaching further into \(q\) — if that control is pointwise in \(t\) (a bound on \(\max_t|\Delta(t;q,a)|\), however strong in \(q\)), then composing it with a Route-D identity by the same trivial route as (P.8) costs a full power of \(N\) again, regardless of the strength in \(q\). What Route D actually needs is a tool that already controls \(\sum_{t\le N}\Delta(t;q,a)^2\) or \(\int_1^N\Delta(t;q,a)^2\,dt\) — an aggregate in \(t\), not only a pointwise-in-\(t\), bound.
This is where Bombieri–Vinogradov and Barban–Davenport–Halberstam enter, and where they fall short of what is needed. Both are aggregate-in-\(q\) statements:
- The Bombieri–Vinogradov theorem states that for every fixed \(A>0\) there is \(B'=B'(A)\) with \(\sum_{q\le Q}\max_{y\le N}\max_{(a,q)=1}|\psi(y;q,a)-y/\phi(q)| \ll_AN/L^A\) for \(Q\le\sqrt N/L^{B'}\) — a range that comfortably contains \(R_0\sim\sqrt N/(3L)\) for suitable \(B'\).
- The Barban–Davenport–Halberstam theorem, in its classical unconditional form, controls \(\sum_{q\le Q}\sum_a^*(\psi(N;q,a)-N/\phi(q))^2\) (a mean square, not a max) on average over \(q\), for \(Q\) at least up to \(\sqrt N\) and, in refined forms this document cannot cite a precise range for from memory alone, considerably further.
Both aggregate over \(q\) (and, for Bombieri–Vinogradov, take a max over \(a\) and over \(t\le N\) rather than summing over \(a\)); neither is stated as an aggregate over \(t\) — neither bounds \(\sum_{t\le N}\Delta(t;q,a)^2\) or its integral analogue for a fixed \(q\), summed or mean-squared over the residues \(a\), the object Route D's identity would actually produce. Whether such a "\(t\)-integrated Barban–Davenport–Halberstam" estimate — something of the shape \[ \sum_{L^B<q\le R_0}\frac{\mu(q)^2}{\phi(q)^2}\sum_a^* \int_1^N\Delta(t;q,a)^2\,dt\ \ll_\epsilon\ N^{2+\epsilon} \tag{P.9} \] (the \(\mu(q)^2/\phi(q)^2\) weight matching how \(P_{q,a}\) itself is weighted, by analogy with (29)-(31)'s conversion of \(\sum_t\Delta(t)^2\) into \(32U_1\)) — is a known unconditional theorem, a known false statement, or simply unexamined, is not something this document can settle: it is not in UPPER_BOUND.md or RESULTS.md, and this attempt has no access to outside literature to search for it. This is the wall this investigation reaches. Naming it this precisely is itself the result of the investigation, not a proof that no such estimate exists.
6. What this changes about RANK3_SCOPE.md's picture, and what it does not
RANK3_SCOPE.md's Route A frames the missing ingredient as "a per-modulus uniformity... valid for individual moduli growing like \(N^{1/2}\)," and notes its only exhibited closing route (Section 7's, at \(q=1\)) assumes RH, calling this "circular with respect to this hunt's own stated target." Section 5 above sharpens this along the sum-reading, \(S_U(B)\), \(S_Z(B)\), specifically:
- The obstruction Route D actually meets is not, in the first instance, a need for RH-strength control at each individual \(q\). It is a need for an aggregate-in-\(q\)-and-\(t\) mean-square estimate, of the same general kind Bombieri–Vinogradov and Barban–Davenport–Halberstam already supply unconditionally (their entire interest is that they give GRH-strength savings on average, without assuming GRH) — just not, so far as this document can determine, in the \(t\)-integrated shape (P.9) needs. This is a different, and on its face less obviously RH-dependent, kind of gap than Route A's per-modulus framing suggests.
- This also answers, provisionally, RANK3_SCOPE.md Route A's open question of whether an averaged-in-\(q\) estimate "would even plug into the arc-by-arc structure": since \(U_Q=\sum_qU_{(q)}\) exactly, an aggregate bound like (P.9) — if it existed and if Route D's identity were carried out to connect it to \(U_{(q)}\), \(Z_{(q)}\) precisely — would plug in directly, with no need for per-arc processing; (18) and (23)'s arc-by-arc form is how the current proof is organized, not a requirement on any proof of a bound on the sum.
- This does not resolve the max-reading, \(\max_qU_{(q)}\): an aggregate bound over \(q\) says nothing about any single summand, so the max-reading reduces to Route A's original, per-modulus, RH-tied question exactly as RANK3_SCOPE.md states it. The sum-reading and the max-reading genuinely differ here.
- Nor does it touch \(q=1\): (P.9)-style aggregation is over many moduli, and gives no separate handle on the single term \(q=1\), which sits outside the range \(\lfloor L^B\rfloor<q\le R_0\) in question here and is RANK3_SCOPE.md's rank 1, unaffected by anything in this document.
7. Would closing this range (if it could be closed) move the reached bound?
By the same argument as RANK3_SCOPE.md Section 4: no, on its own. Rank 1's established bound, \(O_H(N^3L^{-2H})\), remains the weakest link among the three named constraints regardless of what happens in this range, since (23) sums all three rather than taking their minimum. Sections 2-3 above give an additional, concrete reason this range's own contribution, at the lower edge \(q\approx L^B\), sits at exactly the same order, \(O(N^3L^{O(B)})\) — so even if (P.9) or an equivalent were found and proved, the shortfall it would remove is the same shape and, absent progress on rank 1, the same order as what already limits the construction.
8. Summary
For the sum-reading \(S_U(B)\), \(S_Z(B)\): two structurally different routes were checked and both fail throughout \(\lfloor L^B\rfloor<q\le R_0\), for every fixed \(B\), by direct computation from tools already in UPPER_BOUND.md — (P.5)-(P.6) for the currently-used decomposition and large-sieve descent, (P.7) for a direct large-sieve treatment of \(F_N\)'s fourth moment. A third, more promising direction — an aggregate-in-\((q,t)\) mean-square estimate of Barban–Davenport–Halberstam type, (P.9) — is named precisely, is not ruled out by anything derived here, and is not something this document, working without outside literature access, can confirm exists. That is the wall. For the max-reading, \(\max_qU_{(q)}\), \(\max_qZ_{(q)}\): this reduces to RANK3_SCOPE.md's Route A/D per-modulus question and is exactly as tied to an unconditional Siegel-Walfisz-strength uniformity (or RH) as RANK3_SCOPE.md already found, unaffected by anything new here.