against the primary sources it is attributed to
This document checks, against citable primary sources rather than by re-derivation, a specific hypothesis H* that would be needed to push RANK3_MEAN_VALUE_TOOLS.md Section 3's Barban-Davenport-Halberstam route further than that document itself takes it. It does not attempt the Abel-summation splice into \(U_{(q)}\) that a confirmed H* would feed (that is separate work, as the task assigning this document says). It writes nothing about zeros of \(L\)-functions or the Riemann Hypothesis.
A word on sourcing. This attempt has no live network access: WebSearch and WebFetch calls both returned a permission error with no interactive user available to grant it, and no local copy of Montgomery and Vaughan, Multiplicative Number Theory I, Montgomery, Topics in Multiplicative Number Theory (LNM 227), or Davenport, Multiplicative Number Theory exists anywhere under this worktree. This is the same wall the document this task quotes already recorded ("via a URL to course notes ... that this attempt could not fetch"). What follows is therefore sourced from the precise, checkable, standard textbook content of those three books as commonly stated (the same three RANK3_MEAN_VALUE_TOOLS.md Section 1 already cites and whose citations this document reuses rather than re-checks), not from a live fetch of a specific page. Where the exact theorem number or page is not something this attempt could re-verify online, that is flagged explicitly rather than stated as if checked. A later attempt with working network access should confirm exact theorem/page numbers directly against the PDF at the URL UPPER_BOUND.md already cites, or against Davenport or Montgomery's LNM 227 directly, and correct anything below that a live check contradicts.
1. H*, restated exactly
\[ \text{For every fixed }A>0,\text{ uniformly for }1\le Q\le R_0:\qquad \sum_{q\le Q}\ \sum_{\substack{a\bmod q\\(a,q)=1}}\ \sum_{t=1}^N |\Delta_a(t;q)|^2\ \ll_A\ QN^2(\log N)^{C_0-A} \] for some absolute constant \(C_0\), where \(\Delta_a(t;q)=\sum_{n\le t}\big[\Lambda(n)e(na/q)-\mu(q)/\phi(q)\big]\), \(R_0=Q_{\max}/L\) with \(Q_{\max}=\lfloor\sqrt N/3\rfloor\) and \(L=\log N\) (UPPER_BOUND.md Section 6; note UPPER_BOUND.md and RANK3_SCOPE.md both call the arc parameter itself \(Q\), so here "\(Q\)" is H*'s own summation cutoff, distinct from UPPER_BOUND.md's fixed \(Q=\lfloor\sqrt N/3\rfloor\), matching how the task phrasing uses it).
\(\Delta_a(t;q)\) is not RANK3_ROUTE_D.md's or RANK3_MEAN_VALUE_TOOLS.md's \(\Delta(t;q,b)=\psi(t;q,b)-t/\phi(q)\): it is the partial sum, up to \(t\), of \(F_N\)'s integrand at the single rational frequency \(a/q\) minus UPPER_BOUND.md Section 6's own major-arc model density \(P_{q,a}=(\mu(q)/\phi(q))K_N(\alpha-a/q)\) restricted to that frequency (\(\mu(q)/\phi(q)\) is exactly \(P_{q,a}\)'s coefficient there). This is a faithful, if differently indexed, discrepancy quantity for the same approximation the identity actually uses, and Section 2 below relates it precisely to \(\Delta(t;q,b)\).
2. \(\Delta_a(t;q)\) and \(\Delta(t;q,b)\) are the same object, dually
By orthogonality, for \((a,q)=1\), \[ \sum_{n\le t}\Lambda(n)e(na/q) =\sum_{b\bmod q}e(ba/q)\sum_{\substack{n\le t\\ n\equiv b\,(q)}}\Lambda(n) =\sum_{b\bmod q}e(ba/q)\psi(t;q,b), \] and \(\psi(t;q,b)=O(\log q\log t)\) for \((b,q)>1\) (only prime-power divisors of \(q\) contribute), so up to that negligible correction, \[ \sum_{n\le t}\Lambda(n)e(na/q) =\sum_{\substack{b\bmod q\\(b,q)=1}}e(ba/q)\,\psi(t;q,b)+O(\log q\log t). \] Since the Ramanujan sum at \((a,q)=1\) is \(c_q(a)=\mu(q)\), subtracting \(t\mu(q)/\phi(q)=\big(t/\phi(q)\big)\sum_{(b,q)=1}e(ba/q)\) from both sides gives \[ \Delta_a(t;q)=\sum_{\substack{b\bmod q\\(b,q)=1}}e(ba/q)\,\Delta(t;q,b) +O(\log q\log t). \] This is the standard duality RANK3_MEAN_VALUE_TOOLS.md Section 1 already names ("dual, via Gauss sums, to the additive large sieve"), applied here at the level of the partial sum rather than only the full large-sieve inequality; it is bookkeeping, not a new estimate. Summing over \((a,q)=1\) and using orthogonality of \(e(ba/q)\) over the reduced residues (a finite character-sum identity, not the large sieve itself), \[ \sum_{\substack{a\bmod q\\(a,q)=1}}|\Delta_a(t;q)|^2 =\phi(q)\sum_{\substack{b\bmod q\\(b,q)=1}}|\Delta(t;q,b)|^2 +O\big(\phi(q)\log^2q\log^2t\big), \] i.e. H*'s left side, for fixed \(q\), is \(\phi(q)\) times \(D(t,\{q\})\)'s single-modulus summand, up to an error that is negligible at every scale this document or RANK3_MEAN_VALUE_TOOLS.md tracks. So checking H* against a classical statement of (BDH) for \(\Delta(t;q,b)\) is legitimate, not a category error, modulo this standard identification, which this document states once and does not re-derive further.
3. The single-endpoint case: classical (BDH), confirmed, but at one fixed
log power, not "for every \(A\)"
The task's own framing of the single-point special case is exactly the classical theorem: for \(1\le Q\le x\), \[ D(x,Q):=\sum_{q\le Q}\sum_{\substack{b\bmod q\\(b,q)=1}} \big(\psi(x;q,b)-x/\phi(q)\big)^2\ \ll\ Qx\log x \tag{BDH} \] unconditionally (Barban 1966; Davenport and Halberstam 1966), proved via the multiplicative large sieve (Montgomery and Vaughan, Multiplicative Number Theory I, the large-sieve chapter and its Barban-Davenport- Halberstam corollary; Montgomery, Topics in Multiplicative Number Theory, LNM 227, Chapter 4; Davenport, Multiplicative Number Theory, the chapter on the large sieve and primes in arithmetic progressions). This is exactly the form RANK3_MEAN_VALUE_TOOLS.md Section 3 already cites and numerically checks (rank3_bdh_probe.py), and this document adds nothing new to that citation beyond the duality of Section 2. Two range facts matter here, and this document reads them the same way RANK3_MEAN_VALUE_TOOLS.md Section 3 already states them:
- (BDH) as an upper bound holds for the entire range \(1\le Q\le x\), with no lower threshold on \(Q\) and no dependence of the range on any extra parameter -- it is a single valid inequality restricted to fewer terms as \(Q\) shrinks, so it applies at \(Q=R_0\ll\sqrt N\) exactly as it does at \(Q=N\). Range is not an obstruction for the single-endpoint case.
- The sharp asymptotic strengthening is a different, narrower-range statement. Montgomery (1970), and independently Hooley, proved \(D(x,Q)\sim Qx\log Q\) (matching the trivial-looking upper bound as a genuine two-sided asymptotic, not just an upper bound) uniformly for \(x(\log x)^{-A}\le Q\le x\), for every fixed \(A>0\) -- i.e. for \(Q\) within any fixed power of \(\log x\) below \(x\) itself, not for small \(Q\). \(R_0\asymp\sqrt N/(3L)\) is nowhere near this range (it is far below \(x(\log x)^{-A}\) for any fixed \(A\), once \(N\) is large): the "stronger forms much further" the task's own phrasing anticipates reach closer to \(Q=x\), not closer to \(Q=1\). This is consistent with, not contradicting, the previous bullet: the upper bound (BDH) reaches every \(Q\), the sharp asymptotic only reaches large \(Q\). Since H* only needs an upper bound at \(Q\le R_0\), this distinction does not by itself block H* -- but it does mean the "for every \(A\)" language in the sharp form is about the precision of the asymptotic at large \(Q\), not about an arbitrarily strong saving available at every \(Q\) including small \(Q\), which is the reading H* needs and Section 4 addresses.
Reading against H* at the single endpoint. Taking \(t=N\) only (dropping H*'s sum over \(t\)) and comparing to (BDH) via Section 2's duality: (BDH) gives exactly \(QN\log N\), a single fixed power of \(\log N\) -- matching H* with \(C_0=1\) and \(A=0\), i.e. the weakest member of H*'s claimed family. It does not, on its own, give H*'s stronger members: (BDH), as the standard theorem is stated in these three sources, carries no free parameter \(A\) with an arbitrarily strong log saving. The single-endpoint case of H* is confirmed only at \(A=0\) (i.e. as the bound \(\ll QN\log N\), full range \(Q\le R_0\)); the claim "for every fixed \(A>0\)" is a strictly stronger statement that (BDH) as classically stated does not supply, at any \(Q\), let alone uniformly down to \(R_0\).
4. Where an arbitrary-\(A\) log saving would have to come from, and why
it is not available from (BV) either
An arbitrary-\(A\) saving on a mean-square, summed-in-\(q\) quantity is exactly Bombieri-Vinogradov's kind of strength, not Barban-Davenport- Halberstam's. RANK3_MEAN_VALUE_TOOLS.md Section 2 already checked (BV) against this same identity and found: (BV) bounds a sum of max-deviations, not a sum of squares, by \(x(\log x)^{-A}\) for every fixed \(A\), at the cost of a range \(x^{1/2}(\log x)^{-B(A)}\) with \(B\) growing with \(A\); plugging it in reproduces only (SW)'s own log-power strength (\(O_H(N^3L^{-H})\)), "never a saving of any power of \(x\)." That conclusion, reused here rather than re-derived, is exactly why an arbitrary-\(A\) member of H* cannot simply be assembled from (BV) squared, Cauchy-Schwarz'd, or otherwise combined with (BDH)'s shape at a fixed range independent of \(A\): (BV)'s own range shrinks as \(A\) grows (\(B(A)\) increases), while H* asks for the same range \(Q\le R_0\), with no \(A\)-dependence in the range at all, for every \(A\) simultaneously. A hybrid theorem with (BDH)'s square-sum shape and (BV)'s arbitrary-\(A\) saving, valid on a single \(A\)-independent range down to \(Q\le R_0\), is not among the theorems stated in any of the three named books as this document or RANK3_MEAN_VALUE_TOOLS.md read them, and this document did not find one under another name either. This is the same category of gap RANK3_MEAN_VALUE_TOOLS.md Section 5 already surfaces from a different angle (there, for the \(t\)-summed quantity specifically); Section 5 below gives the \(t\)-summed version of the same gap.
5. The \(t\)-sum: available at order \(QN^2\log N\) (i.e. H* with
\(A=0\)), not with any further saving found
UPPER_BOUND.md's own remark, quoted in the task, that "the version with a maximum over \(t\) follows from the usual theorem by treating \(t\le N/L^D\) trivially and applying it above that point," is stated for (SW), whose content is an error term \(O_{B,H}(NL^{-H})\) against a main term \(t/\phi(q)\) of full size \(t\): trivially bounding \(|\Delta(t;q,b)|\le t\) for \(t\le N/L^D\) costs at most \(N/L^D\), which is already smaller than (SW)'s own claimed saving once \(D\) is chosen large enough relative to \(H\) -- the trivial range is absorbed into the error budget the theorem already has.
This trick does not transfer to (BDH)'s quantity. (BDH)'s content, \(D(t,Q)\ll Qt\log t\), is not an error term dominated by a separate larger main term; it is the theorem's leading-order content, and it scales with \(t\) itself (RANK3_MEAN_VALUE_TOOLS.md Section 3: "a full power of both \(Q\) and \(x\) saved... down to every \(Q\le x\)"). Truncating \(t\le N/L^D\) "trivially" means falling back to the trivial per-\(t\) bound (order \(Q^2t^2\), RANK3_MEAN_VALUE_TOOLS.md Section 3's own accounting), which is far larger than \(D(t,Q)\) itself at that same \(t\), not smaller -- there is no saving to fall back into. Consequently no dyadic-dissection bridge of the (SW) kind rescues a \(t\)-summed (BDH) at better than the crude bound obtained by applying (BDH) separately at each \(t\le N\) and summing: \[ \sum_{t=1}^N D(t,Q)\ \ll\ \sum_{t=1}^N Qt\log t\ \ll\ QN^2\log N, \] exactly RANK3_MEAN_VALUE_TOOLS.md's own (E1), independently re-derived here from the structural reason the (SW)-style shortcut fails rather than assumed. This is H* with \(C_0=1\), \(A=0\) again -- the same, weakest member of H*'s family, and this document finds no route in the three named sources, nor a structural bridge from (SW)'s own trick, to any stronger member (\(A>0\)) of the \(t\)-summed claim either.
6. Verdict
Confirmed, precisely: the single-endpoint, \(A=0\) member of H*, \(\sum_{q\le Q}\sum_a^*|\Delta_a(N;q)|^2\ll QN\log N\), uniformly for the entire range \(1\le Q\le N\) (so in particular \(Q\le R_0\)) -- this is classical (BDH), via Section 2's duality, and is the content RANK3_MEAN_VALUE_TOOLS.md Section 3 already cites and measures.
Confirmed, at the same weakest member only: the \(t\)-summed extension, at order \(QN^2\log N\) (H* with \(C_0=1\), \(A=0\)), via the crude per-\(t\) sum of Section 5 -- matching, not exceeding, RANK3_MEAN_VALUE_TOOLS.md's own (E1)/(E2).
Not found, and not supported by (BDH) or (BV) as stated in Montgomery and Vaughan, Montgomery's LNM 227, or Davenport: the "for every fixed \(A>0\)" clause of H*, at either the single-endpoint or the \(t\)-summed form, uniformly on the single range \(Q\le R_0\). Section 3 shows (BDH)'s sharp, arbitrary-precision asymptotic form reaches the wrong end of the \(Q\)-range (large \(Q\), not small); Section 4 shows (BV)'s arbitrary-\(A\) strength comes with an \(A\)-shrinking range, incompatible with H*'s fixed range; Section 5 shows the natural (SW)-style bridge for extending to a \(t\)-sum does not transfer to (BDH)'s shape. What is missing, stated as precisely as this document can: a mean-square (not max-deviation) analogue of Bombieri-Vinogradov -- an arbitrary-\(A\), \(A\)-independent-range, \(t\)-integrated Barban-Davenport-Halberstam theorem -- and this document does not find one named in the three sources it checked, nor construct one. This is a refinement of, not a departure from, RANK3_MEAN_VALUE_TOOLS.md Section 8's own list of missing ingredients (its item (i)); this document's contribution is pinning exactly which two of H*'s three strengthenings (arbitrary \(A\), and the \(t\)-sum) are and are not covered by the named primary sources, and why, rather than treating H* as a single yes/no question.
Even if the missing hybrid theorem existed and gave the strongest member of H* at every fixed \(A\): plugged into RANK3_ROUTE_D.md (D11) at \(Q=R_0\), it would give \(U_{(q)}\)'s first term order \(R_0N^2(\log N)^{C_0-A}\ \asymp\ N^{5/2}(\log N)^{-1-A}\) for every fixed \(A\) -- an improvement over RANK3_MEAN_VALUE_TOOLS.md Section 5's own \(N^{5/2}\) by an arbitrary power of \(\log N\), but, by RANK3_MEAN_VALUE_TOOLS.md Section 2's own reasoning applied here verbatim, never by any power of \(N\): a log saving, however large but fixed, cannot on its own close the full power \(N^{1/2}\) gap to \(N^{2+\epsilon}\). So confirming the missing piece of H* would sharpen, but not by itself close, the same door RANK3_MEAN_VALUE_TOOLS.md Section 5 already prices -- consistent with that document's own conclusion and RANK3_SCOPE.md Section 3's, that no route named so far is costed to completion. This point is recorded here because it bears on whether pursuing H*'s missing piece is worth the next attempt's time, not as the substance of the verify/refute task itself, which Section 6's first three paragraphs above answer.
This document is a citation check (Sections 1, 3, 4) plus one finite, standard duality identity (Section 2) plus a structural argument for why a specific bridging trick does not transfer (Section 5); it assumes and establishes nothing about zeros of \(L\)-functions or the Riemann Hypothesis.