not close rank 3's polynomial range either
This document investigates, as RANK3_SCOPE.md Section 2 asks, whether some unconditional tool other than the three named in UPPER_BOUND.md -- the Siegel-Walfisz major-arc approximation (17)/(SW), Vaughan's exponential-sum estimate (V), and the additive large sieve (LS) used dyadically in (24)-(27) -- could bound the mixed and fourth moments \(U_{(q)}\), \(Z_{(q)}\) for denominators in the polynomial-in-\(N\) range \(\lfloor L^B\rfloor<q\le R_0\) (\(L=\log N\), \(Q=\lfloor\sqrt N/3\rfloor\), \(R_0=Q/L\)), where RANK3_SCOPE.md shows all three named tools give only a trivial-order bound. RANK3_SCOPE.md's Section 3 names and prices four routes (A-D), all built from the same three tools; this document looks at a fifth: mean-value theorems for primes in arithmetic progressions -- Bombieri-Vinogradov and Barban-Davenport-Halberstam -- which rest on the multiplicative (character) large sieve rather than the additive (Farey-fraction) large sieve (LS) that (24)-(27) already uses. This is "a different large-sieve weighting" in the sense RANK3_SCOPE.md's task asks about: same underlying large-sieve mechanism, different indexing (by Dirichlet character rather than by rational point), and, as shown below, a genuinely different degeneration profile at small \(q\). The finding is negative: this route names a real, previously-unnamed candidate, gets further than routes A-D on the shape mismatch, but stops short of \(N^{2+\epsilon}\) by exactly one unresolved factor of \(N\), which is named precisely in Section 5, plus the separate transfer-step obstruction RANK3_ROUTE_D.md Section 6 already identifies, which this route does not touch either. \(Z_{(q)}\) fares worse: no candidate tool for it is found here at all (Section 6).
Notation throughout is UPPER_BOUND.md's and RANK3_ROUTE_D.md's: \(F_N,K_N, \Lambda,\psi,\mu,\phi\) as in UPPER_BOUND.md Section 1; \(P_{q,a},R_{q,a}, U_{(q)},Z_{(q)}\) as in UPPER_BOUND.md Section 6 and RANK3_SCOPE.md Section 1; \(\Delta(t;q,b)=\psi(t;q,b)-t/\phi(q)\) and \(T(q,b)\) as in RANK3_ROUTE_D.md (D5), (D7), (D11). This document assumes RANK3_ROUTE_D.md as already established and does not re-derive its identities.
1. The candidate: mean-value theorems, and why they are a different tool
Write \(\Delta(x;q,a)=\psi(x;q,a)-x/\phi(q)\) for the classical (not \(K_N\)-convolved) prime-counting discrepancy in an arithmetic progression. Two classical unconditional theorems bound this, on average over \(q\), far beyond \(q\le L^B\):
Bombieri-Vinogradov (Bombieri 1965; A. I. Vinogradov 1965; see Davenport, Multiplicative Number Theory, Chapter 28, or Montgomery and Vaughan, Multiplicative Number Theory I, Chapter 17). For every \(A>0\) there is \(B=B(A)\) such that \[ \sum_{q\le x^{1/2}(\log x)^{-B}}\ \max_{y\le x}\ \max_{(a,q)=1} \big|\psi(y;q,a)-y/\phi(q)\big|\ \ll_A\ x(\log x)^{-A}. \tag{BV} \]
Barban-Davenport-Halberstam (Barban 1966; Davenport and Halberstam 1966; sharp asymptotic form Montgomery 1970, and independently Hooley; see Montgomery, Topics in Multiplicative Number Theory, Springer Lecture Notes in Mathematics 227, Chapter 4, and Montgomery and Vaughan, Multiplicative Number Theory I, Chapter 17 and its notes). Uniformly for \(1\le Q\le x\), \[ D(x,Q):=\sum_{q\le Q}\ \sum_{\substack{a\bmod q\\(a,q)=1}} \big(\psi(x;q,a)-x/\phi(q)\big)^2\ \ll\ Qx\log x. \tag{BDH} \]
Neither is named in UPPER_BOUND.md or RESULTS.md. Both are proved via the multiplicative large sieve -- the large sieve inequality for Dirichlet characters, sum_{q\le Q}\sum_{\chi\bmod q}|\sum_n a_n\chi(n)|^2\le(N+Q^2) \sum_n|a_n|^2 (Montgomery and Vaughan, Multiplicative Number Theory I, Theorem 6.7, or Iwaniec and Kowalski, Analytic Number Theory, Theorem 7.13) -- which is dual, via Gauss sums, to the additive large sieve (LS) UPPER_BOUND.md already cites, but is applied differently: (24)-(27) applies (LS) to points \(a/q\) restricted to a single dyadic block \(R<q\le2R\), one block at a time, then sums the blocks; (BV) and (BDH) apply the character large sieve to all moduli \(q\le Q\) simultaneously, in one inequality, then subtract off the low-\(q\) part.
Why this changes the small-\(q\) degeneration. RANK3_SCOPE.md Section 2 identifies exactly where (24)-(27) fails: the per-block estimate (25), applied to the smallest block \(R=1\) (i.e. \(q=2\)), gives \(U_{q=2}\ll w(Q)^2N^3L\), because the large-sieve ceiling \((N-1+4R^2)d_N\) in (24) does not shrink with \(R\) -- for one modulus with few points, the large sieve saves nothing over the trivial bound on that block alone. The same objection applies, term for term, to a dyadic-block version of the multiplicative large sieve: restricting the character large sieve to \(R<q\le2R\) and dividing by \(\phi(q)\sim R\) to normalize (needed to extract \(D(x,Q)\)'s per-\(q\) sum \(\sum_a(\Delta)^2=(1/\phi(q))\sum_\chi |\psi(x,\chi)|^2+O(\cdots)\) from a bound on the unweighted \(\sum_\chi |\psi(x,\chi)|^2\)) reproduces the same failure at \(R=O(1)\): a single modulus contributes nothing to a large-sieve saving on its own. (BV) and (BDH), as actually proved, do not go through this block-by-block route at all -- they apply the character large sieve to the entire range \(q\le Q\) in one inequality and are stated, and provably hold, uniformly down to \(Q=O(1)\) (trivially, since restricting a valid inequality to a sub-range of nonnegative terms cannot increase it). This is the substantive difference from routes A-D: it is not a re-application of (LS) in a new place, it is a use of the large-sieve mechanism that structurally avoids the specific dyadic-block degeneration RANK3_SCOPE.md Section 2 diagnoses. Section 2 below checks this on real data.
2. Bombieri-Vinogradov is the wrong strength, not the wrong range
\(R_0=Q/L\) with \(Q=\lfloor\sqrt N/3\rfloor\), so \(R_0\asymp\sqrt N/(3L)\). (BV)'s range, \(x^{1/2}(\log x)^{-B}\) for \(B=B(A)\), comfortably contains \(R_0\) for suitable fixed \(A\) (any fixed \(B\) exceeds a fixed power of \(\log N\), and \(R_0\) is \(\sqrt N\) divided by one power of \(L\), so for a large enough choice of \(A\) -- hence \(B\) -- (BV)'s range reaches past \(R_0\)). Range is not the obstruction here. Strength is: (BV) bounds a sum (not a sum of squares) of the max-over-\(y\), max-over-\(a\) discrepancy, summed over up to \(x^{1/2-o(1)}\) moduli, by \(x(\log x)^{-A}\) for every fixed \(A\) -- an arbitrary saving of a power of \(\log x\), never a saving of any power of \(x\). This is exactly the strength (SW) already has at \(q\le L^B\): both are "Siegel-Walfisz-type" statements, and (BV) is provably no stronger in kind, only wider in averaged range. Plugging (BV) into the identity in place of the individual-\(q\) uniformity Route A needs would therefore, in the best case, extend an (17)-style major-arc approximation across the summed range \(q\le R_0\) at (SW)'s own strength: a bound of shape \(O_H(N^3L^{-H'})\) for every fixed \(H\) (matching rank 1's own established order, RANK3_SCOPE.md Section 1), not \(O_\epsilon(N^{2+\epsilon})\). RANK3_SCOPE.md Section 4 already shows that rank 1's log-power bound is the current bottleneck of (23), and that closing another door to the same log-power strength does not move the total order. So even a fully worked out application of (BV) to rank 3's mixed moment would reproduce, not beat, an obstruction the text already has. This is a clean, low-cost conclusion that needs no numerical check: it follows from (BV)'s own stated strength, which this document cites rather than measures. (The Elliott-Halberstam conjecture, extending (BV)'s range to \(Q\le x^{1-\epsilon}\), changes nothing about this point -- it is conditional, not admissible here, and, in any case, only widens the range, not the log-vs-power-saving character of the bound.)
3. Barban-Davenport-Halberstam is the right shape and the right strength
(BDH) is a sum of squares, matching \(T(q,b)\)'s structure in RANK3_ROUTE_D.md (D5) far better than (BV) does, and its unconditional strength is genuinely power-type: \(D(x,Q)\ll Qx\log x\) against a trivial per-term bound of \(4x^2\) (Chebyshev: \(|\psi(x;q,a)|,x/\phi(q)\le\psi(x) \ll x\)) summed over \(\sum_{q\le Q}\phi(q)\asymp Q^2\) terms, i.e. a trivial ceiling of order \(Q^2x^2\). (BDH)'s \(Qx\log x\) is smaller than this trivial ceiling by a factor of order \(Qx/\log x\) -- a full power of both \(Q\) and \(x\) saved, unconditionally, down to every \(Q\le x\) including small \(Q\). This is the kind of saving rank 3 needs and (25)- (26) cannot supply at small \(q\).
Numerical check. rank3_bdh_probe.py (results in results_rank3_bdh_probe.json) measures \(D(N,Q)=\sum_{q\le Q}\sum_{a\bmod q}^*(\psi(N;q,a)-N/\phi(q))^2\) directly from the true von Mangoldt function (prime-power enumeration, no sampling), two ways.
Fixed \(Q=30\), \(N\) from \(2\times10^4\) to \(8\times10^5\). \(D(N,Q)/N\) stays in the narrow band \(10.8\) to \(13.7\) with no systematic trend across a 40-fold range of \(N\); \(D(N,Q)/(N\log N)\) stays in \(0.94\) to \(1.21\). This cleanly confirms the \(N\)-dependence of \(D(N,Q)\) is linear in \(N\), matching (BDH)'s \(N\)-power exactly -- this is the power that matters most for Section 5's scaling argument, and it is confirmed directly rather than assumed.
Fixed \(N=150000\), \(Q\) from \(10\) to \(500\). \(D(N,Q)/(QN)\) rises from \(0.08\) to \(2.25\) -- a 28-fold increase against a 50-fold increase in \(Q\), i.e. faster than the \(\log(500)/\log(10)\approx2.7\)-fold rise a pure \(Q\log Q\) shape would predict. This experiment, alone, cannot tell a pre-asymptotic regime (a known feature of mean-value theorems: the displayed leading term needs \(Q\) large enough relative to lower-order terms before it dominates) from a genuinely steeper \(Q\)-dependence at these particular finite parameters, and is reported as measured, not fit to either shape; \(Q=500\) is close to \(\sqrt N=387\), where the large-sieve ceiling \((N+Q^2)\) itself crosses from \(N\)-dominated to \(Q^2\)-dominated, which is a plausible source of the acceleration seen here and is a reason for caution about extrapolating this \(Q\)-range number to \(Q=R_0\ll\sqrt N\). What both sub-experiments confirm robustly: \(D(N,Q)\) is seven to eight orders of magnitude below the trivial ceiling \(Q^2N^2\) at every checkpoint (ratio_measured_over_trivial in the JSON, ranging only from \(9.3\times10^{-8}\) down to \(3.0\times10^{-8}\) across the whole 50-fold range of \(Q\) -- itself varying by less than a factor of \(3\), meaning the ratio to the trivial \(Q^2N^2\) bound does not run away either), i.e. a genuine, large, measurable saving is present even at the small-to-moderate \(q\) where (25)-(26) gives none.
Section 5 below uses only the classical cited form (BDH), \(D(x,Q)\ll Qx \log x\), for the scaling argument, and only cross-checks its \(N\)-power against the clean measurement above; it does not lean on this document's own uncertain read of the \(Q\)-power.
4. Where (BDH) would have to plug in, and why (D11)'s transfer term is untouched
RANK3_ROUTE_D.md (D11) already reduces \(U_{(q)}\) (up to the \(R^{(2)}\) correction, controlled there) to \[ U_{(q)}\ \le\ 2\mu(q)^2\sum_{b\bmod q}^*T(q,b)\ +\ O\!\Big(\frac{q^2N^3L} {Q^2\phi(q)}\Big), \] with \(T(q,b)=\sum_{t=1}^N\Delta(t;q,b)^2+\sum_{t=1}^{N-1} [\Delta(N;q,b)-\Delta(t;q,b)]^2\) (RANK3_ROUTE_D.md (D5)). (BDH) is a candidate for the first term. It is not a candidate for the second: the transfer term is already shown, in RANK3_ROUTE_D.md Section 6, to cost \(O(N^3)\) once summed over \(2\le q\le R_0\) regardless of how well \(\sum_b^*T(q,b)\) is controlled -- "the obstruction is the transfer step itself, not the main term." Nothing in this document changes that; a bound on \(T(q,b)\) from (BDH) would still need RANK3_ROUTE_D.md Section 6's own missing ingredient (a dyadic, non-crude transfer argument) before it could help \(U_{(q)}\) at all. Section 5 below therefore prices only the first term, on the understanding -- already on record in RANK3_ROUTE_D.md -- that a second, independent obstruction remains even if it fully succeeds.
5. Pricing (BDH) against the first term: short by exactly one factor of \(N\)
(BDH)'s object, \(D(t,Q)=\sum_{q\le Q}\sum_b^\Delta(t;q,b)^2\), is a single-endpoint (in \(t\)) quantity. \(T(q,b)\), summed over \(q,b\), needs \(\sum_{q\le Q}\sum_b^\sum_{t=1}^N\Delta(t;q,b)^2\) -- the same quantity summed again over every \(t\) from \(1\) to \(N\), not evaluated once at \(t=N\). (BDH) is proved uniformly in the endpoint (Section 1's citations state it for \(x\le\) the stated range, and the standard proof, via the character large sieve applied to \(\Lambda(n)\mathbf 1_{n\le t}\), holds at every integer endpoint \(t\) on the same terms as at \(N\)), so it may be applied separately at each \(t\le N\): \[ D(t,Q)\ll Qt\log t\le QN\log N\qquad\text{for every }1\le t\le N, \] and summing this bound over the \(N\) values of \(t\) -- the crude combination this document uses to bridge the endpoint-vs-full-sum gap, clearly separate from anything (BDH) itself asserts -- gives \[ \sum_{t=1}^N D(t,Q)\ \ll\ QN^2\log N,\qquad\text{hence}\qquad \sum_{q\le Q}\sum_b^*T(q,b)\ \ll\ QN^2\log N \tag{E1} \] (the second summand of \(T(q,b)\) is controlled the same way, by \((x-y)^2\le2x^2+2y^2\) applied to \([\Delta(N;q,b)-\Delta(t;q,b)]^2\), without changing the order). At \(Q=R_0=\lfloor\sqrt N/3\rfloor/L\asymp \sqrt N/(3L)\), and using \(2\mu(q)^2=O(1)\) from (D11) (the proved, cancellation-free weight RANK3_ROUTE_D.md Section 5 establishes on \(\sum_b^*T(q,b)\); the smaller, unproven \(\mu(q)^2/\phi(q)\) or \(\mu(q)^2/\phi(q)^2\) weights that Section 5 discusses would only improve this, not worsen it, so (E1) is used with its own weight for an upper bound): \[ \sum_{L^B<q\le R_0}U_{(q)}^{(\text{first term})}\ \ll\ R_0N^2\log N \ \asymp\ \frac{\sqrt N}{3L}\cdot N^2\cdot L\ =\ \frac{N^{5/2}}3. \tag{E2} \] The \(\log N\) from (BDH) and the \(1/L\) from \(R_0\)'s own definition exactly cancel (\(L=\log N\)), leaving a bare \(N^{5/2}\) with no residual log saving at all in this crude combination -- a bookkeeping accident of how coarsely (E1) sums over \(t\), not a claim that this route reaches \(N^{5/2}\) without logs more precisely than CHHL's own conditional statement does; this document does not track log powers past the point where they stop affecting the conclusion below, and (E2)'s log-free appearance should be read as "this crude argument's logs happen to cancel," not as a sharper log power than a more careful argument would give.
The comparison that matters. \(N^{5/2}\) is precisely the order of CHHL's own conditional benchmark, \(E(N)\ll N^{5/2}L^c\) under GRH for Dirichlet \(L\)-functions (UPPER_BOUND.md Section 1, item 1) -- the same power of \(N\), reached here through an entirely different, unconditional route (the character large sieve, applied optimally and without any individual-\(q\) uniformity assumption), landing at exactly the exponent that assuming GRH already gives. That is a meaningful, if negative, finding: the strongest unconditional averaged second-moment tool available for this identity does not beat the GRH-conditional order it is trying to reach unconditionally; it matches it. The gap to the actual target, \(N^{2+\epsilon}\), is one full power of \(N^{1/2}\) -- exactly the power lost in (E1)'s crude step of summing \(N\) copies of the single-endpoint bound. Closing that gap would need an integrated mean-value theorem -- a genuine bound on \(\sum_{q\le Q}\sum_b^* \sum_{t\le N}\Delta(t;q,b)^2\) of order \(QN^{1+\epsilon}\), not the \(QN^2\log N\) that summing the endpoint bound over \(t\) naively gives -- and this document does not find such a theorem named or provable from (BV)/(BDH) as customarily stated. Whether one exists in the literature under some other name (a "mean square of \(\psi(t;q,a)-t/\phi(q)\) integrated over \(t\)," in the spirit of the Barban-Vehov / Motohashi weighted sieve literature, or a large-sieve density theorem for zeros of \(L(s,\chi)\) averaged over \(\chi\) and over height simultaneously) is exactly the open question this document surfaces and does not resolve.
6. \(Z_{(q)}\): no candidate found
(BDH) is a second-moment tool. \(Z_{(q)}=\sum_a^*\int_{\rm arc}|R_{q,a}|^4\) is a fourth moment, and RANK3_ROUTE_D.md Section 7 already shows it does not even reduce to \(\Delta(t;q,b)\)-type quantities in the first place -- the identity's quadruple additive convolution \(Y(b_1,b_2,b_3,b_4)\) has no telescoping structure, a structural obstruction independent of what estimate is available for \(\Delta\). A fourth-moment analogue of (BDH) -- bounding \(\sum_{q\le Q}\sum_a^*|\psi(x;q,a)-x/\phi(q)|^4\), say -- is not part of the two standard theorems cited in Section 1, and this document did not find, or attempt to derive, such a statement with any confidence; it is named here as a missing ingredient, not supplied. Independently, even granting some fourth-moment mean-value bound, RANK3_ROUTE_D.md Section 7's second point stands unaffected: \(Z_{(q)}\)'s integrand carries no \(|P_{q,a}|^2\) factor, so there is no arc-transfer argument of the kind Section 4 uses for \(U_{(q)}\), and a full-circle mean-value bound would not, on its own, bound the arc-restricted \(Z_{(q)}\). So this document finds no candidate tool for \(Z_{(q)}\) over the polynomial range, not even one that falls short the way Section 5's does for \(U_{(q)}\).
7. A different splitting of the arcs by denominator
RANK3_SCOPE.md's task also asks about a different splitting of the arcs by denominator (e.g., by smoothness/friability of \(q\), by \(\omega(q)\), or by largest prime factor) rather than a different estimate. Section 1 above is the reason this does not obviously help: (BV) and (BDH) are already whole-range statements over \(q\le Q\) that do not degrade based on how the range is internally partitioned -- restricting either theorem to any measurable sub-collection of \(q\in(L^B,R_0]\) (friable \(q\), \(q\) with few prime factors, or any other structured subset) only removes nonnegative terms from the same valid inequality, so it cannot produce a bound better in order than applying the theorem to the whole range at once, which Section 5 already prices. A finer split could in principle help if it exploited some extra structure specific to a sub-class of \(q\) (for instance, if \(\Delta(t;q,b)\) were provably smaller, individually, for smooth \(q\) than the general uniform-in-\(q\) case) -- but that would be a return to Route A's individual-modulus uniformity question (RANK3_SCOPE.md Section 3), not a new averaged tool, and RANK3_SCOPE.md already shows no unconditional source for that beyond \(q\le L^B\). This document does not find that any denominator-splitting changes Section 5's conclusion.
8. Where this leaves rank 3's polynomial range
A genuinely different, previously unnamed unconditional tool exists for this range -- the multiplicative large sieve and its consequences (BV), (BDH) -- and it behaves qualitatively better than the additive-large-sieve dyadic-block argument (24)-(27) at small \(q\), both by citation (Section
- and by direct measurement (Section 3,
results_rank3_bdh_probe.json).
But:
- (BV) is the wrong strength (log-saving only, matching rank 1's own bottleneck, not a power-of-\(N\) saving) -- Section 2, no numerical check needed.
- (BDH) is the right shape and strength for the piece of \(U_{(q)}\) it could reach, but only reaches \(N^{5/2}\) order once bridged (by the one crude, clearly-labeled step in Section 5) from a single-endpoint theorem to the full-\(t\)-sum quantity \(T(q,b)\) the identity actually needs -- exactly matching, not beating, CHHL's GRH-conditional benchmark, and falling short of \(N^{2+\epsilon}\) by one power of \(N^{1/2}\).
- Even if that gap were closed, RANK3_ROUTE_D.md Section 6's transfer-step obstruction -- an independent \(O(N^3)\) cost, untouched by any bound on \(T(q,b)\) -- would remain for \(U_{(q)}\).
- No candidate at all is found here for \(Z_{(q)}\) (Section 6).
- Splitting the arcs by denominator differently does not change any of this (Section 7).
What would be needed, stated as precisely as this document can state it: (i) an integrated-in-\(t\), summed-in-\(q\) mean-value theorem for \(\sum_{q\le Q}\sum_b^*\sum_{t\le N}\Delta(t;q,b)^2\) at order \(QN^{1+\epsilon}\), not the \(QN^2\log N\) the customary (BDH) gives once naively summed over \(t\); (ii) independently, the non-crude arc-transfer argument RANK3_ROUTE_D.md Section 6 already names as missing; (iii) any fourth-moment mean-value analogue of (BDH) for \(Z_{(q)}\), or some other route to it entirely, given the structural (not just strength) obstruction RANK3_ROUTE_D.md Section 7 identifies. None of the three is supplied here. This is consistent with, and adds one further specific candidate and its specific failure mode to, RANK3_SCOPE.md Section 3's conclusion that no route among those it names is costed to completion by the source documents.
This document is citation of standard theorems (Section 1), a scaling argument built on them with one clearly labeled crude step (Section 5), and a direct numerical measurement of the ingredient second moment (Section 3); it assumes and establishes nothing about zeros of \(L\)-functions or the Riemann Hypothesis.