Barban-Davenport-Halberstam claim against the literature -- and against
elementary consequences of the classical theorem this hunt already cites
This document answers the task assigned to it: search the unconditional analytic number theory literature for a "t-integrated" analogue of the Barban-Davenport-Halberstam (BDH) / Bombieri-Vinogradov (BV) theorems, in the exact shape and \(q\)-range RANK3_POLYRANGE.md Section 5 needs, or report precisely that none is found. This attempt's worktree does not contain RANK3_POLYRANGE.md -- the same situation RANK3_ROUTE_D.md §5 already records for its own assignment -- so the statement below is taken verbatim from this document's own task description, not read from that file directly; whoever next holds both should reconcile them.
1. The statement, restated exactly
For \(L=\log N\), \(Q=\lfloor\sqrt N/3\rfloor\), \(R_0=Q/L\), and any fixed \(B>0\), is it a known unconditional theorem that \[ S(N,B):=\sum_{q=\lfloor L^B\rfloor+1}^{R_0}\frac{\mu(q)^2}{\phi(q)^2} \sum_{\substack{a\bmod q\\(a,q)=1}}\int_1^N\big(\psi(t;q,a)-t/\phi(q)\big)^2\,dt =O_\epsilon(N^{2+\epsilon}) \] for every \(\epsilon>0\)? \(\psi(t;q,a)=\sum_{n\le t,\,n\equiv a(q)}\Lambda(n)\). Since \(\psi(t;q,a)\) is a step function constant on each unit interval \([t,t{+}1)\), \(\int_1^N(\cdots)^2\,dt\) and \(\sum_{t=1}^N(\cdots)^2\) agree up to an \(O(1)\)-per-\(t\) discretization difference that does not change any order tracked here; this document, and the script accompanying it, work with the discrete sum, matching the convention RANK3_ROUTE_D.md (D5) and RANK3_MEAN_VALUE_TOOLS.md already use for the same object, \(T(q,b)=\sum_t\Delta(t;q,b)^2+(\text{telescoping term})\).
2. The literature search: no live access, same wall as this hunt's other citation checks
WebSearch and WebFetch were both attempted from this worktree and both returned "you haven't granted it yet" with no interactive user able to grant it -- the identical wall RANK3_BDH_VERIFY.md §"A word on sourcing" already recorded, and the one RANK3_POLYRANGE.md itself is quoted (in this document's own assignment) as having hit first, via the personal.science.psu.edu course-notes URL UPPER_BOUND.md cites. No PDF or local copy of Montgomery and Vaughan (Multiplicative Number Theory I), Montgomery (Topics in Multiplicative Number Theory, LNM 227), Davenport (Multiplicative Number Theory), or any paper by Motohashi, Friedlander- Iwaniec, Fouvry, or Bombieri-Friedlander-Iwaniec exists under this worktree. What follows is therefore sourced from the standard, checkable content of those theorems as this hunt's own sibling documents already cite them, plus this document's general knowledge of the named literature strands -- not from a live fetch -- exactly the caveat RANK3_BDH_VERIFY.md states for itself, reused here rather than re-argued.
Reading the named literature strands against the exact shape asked for. None of (BV), (BDH), Motohashi's mean-value refinements, the Friedlander-Iwaniec/Fouvry/Bombieri-Friedlander-Iwaniec line on primes in arithmetic progressions to smooth or well-factorable moduli, or the "variance of primes in short intervals/arithmetic progressions" literature (Hooley, Goldston-Montgomery, Keating-Rudnick and its number-field-only analogue) is, to this document's knowledge, stated as an integral- or sum-over-\(t\) (rather than single-endpoint or max-over-\(t\)) second moment. Every version of (BDH) this document is aware of -- including the sharp asymptotic strengthening by Montgomery (1970) and Hooley -- is stated at a single endpoint \(x\) (or, in some refinements, with a \(\max_{y\le x}\) outside the \(q\)-sum, which is Bombieri-Vinogradov's shape, not a sum or integral over the endpoint). The Motohashi/Friedlander-Iwaniec/Fouvry/BFI line extends (BV)'s range (to well-factorable or smooth moduli near \(x^{4/7}\), \(x^{1/2+\theta}\) for small \(\theta\), etc.) and BFI's own further work extends the level of distribution in the Type II/III sums used for bounded gaps between primes, but none of these, as far as this document is aware, changes (BV)'s or (BDH)'s endpoint structure to an integral over \(t\). The Barban-Vehov/Motohashi weighted sieve literature (which RANK3_MEAN_VALUE_TOOLS.md §5 names as a place such a statement "might" live, without confirming one) concerns weighted second moments in \(q\) at a fixed endpoint, still, to this document's knowledge -- it is a different kind of weighting (a sieve weight on \(q\), not an integral over \(t\)) and this document does not find, or recall, a genuine \(t\)-integrated member of that family either. No theorem of the literal \(t\)-integrated shape asked for here is found, under any of the names the task lists, consistent with (and no stronger a finding than) what RANK3_BDH_VERIFY.md §6 already concluded for the closely related, harder H* question, and what RANK3_MEAN_VALUE_TOOLS.md §5 already flagged as an open question it does not resolve.
This is a negative literature finding, reported as precisely as this document's access allows: not "no such theorem exists" (this document cannot prove a negative over literature it cannot search live), but "this document, working from the standard content of the named sources and no live search, does not find one, and none of the three sibling documents in this hunt that already checked adjacent statements against the same sources (RANK3_BDH_VERIFY.md, RANK3_MEAN_VALUE_TOOLS.md) found one either."
3. What the literature search alone does not settle: \(S(N,B)\) is provable anyway
Section 2's negative finding is not the end of the story for this specific statement, because \(S(N,B)\)'s extra factor \(1/\phi(q)^2\) -- absent from every version of H* RANK3_BDH_VERIFY.md checked, and absent from (BDH) itself -- is strong enough that the classical, single-endpoint (BDH), summed crudely over \(t\) exactly the way RANK3_MEAN_VALUE_TOOLS.md §5 already does to get its own (E1), and then summed over \(q\) by an elementary partial summation (Abel summation) against the decaying weight \(\mu(q)^2/\phi(q)^2\), already gives \(S(N,B)=O_\epsilon(N^{2+\epsilon})\) for every fixed \(B>0\) -- without needing any \(t\)-integrated theorem beyond what this hunt already cites. This is a derivation this document performs itself, not a citation, and it is reported as such.
Setup. Write \(f(q):=\sum_{a\bmod q}^\sum_{t=1}^N\Delta(t;q,a)^2\ge0\) (the discrete form of \(S(N,B)\)'s inner double sum, dropping the \(1/\phi(q)^2\) weight for the moment), and \(A:=\lfloor L^B\rfloor\). For \(A<Q\le R_0\), define the running sum \(F(Q):=\sum_{q=A+1}^Q f(q)\). Since \(f(q)=\sum_t d(t,q)\) with \(d(t,q):=\sum_a^\Delta(t;q,a)^2\) the per-\(q\) summand of (BDH)'s own \(D(t,Q)=\sum_{q\le Q}d(t,q)\), \[ F(Q)=\sum_{t=1}^N\big[D(t,Q)-D(t,A)\big]\ \le\ \sum_{t=1}^N D(t,Q) \ \ll\ \sum_{t=1}^N Qt\log t\ \ll\ QN^2\log N, \] using classical (BDH), \(D(t,Q)\ll Qt\log t\) uniformly for \(1\le Q\le t\) (Barban 1966; Davenport-Halberstam 1966; see the sources cited in Section 1 of RANK3_MEAN_VALUE_TOOLS.md and RANK3_BDH_VERIFY.md), applied separately at each integer \(t\le N\) and summed -- exactly the crude step RANK3_MEAN_VALUE_TOOLS.md §5 already uses to get its own (E1), reused here, not re-derived. (For the small range \(t<Q\), where (BDH)'s stated range \(Q\le x\) does not literally cover \(D(t,Q)\), a direct trivial estimate -- each term \(\Delta(t;q,a)^2=O((\log t)^2)\) since at most one integer \(n\le t<q\) can lie in a given residue class -- gives \(D(t,Q)=O(Q^2\log^2t)\) there, smaller order than \(Qt\log t\) once summed over the negligibly short range \(t<Q\le R_0\ll\sqrt N\); this document does not belabor this boundary case further, as it does not change the order below.) So \(F(Q)\ll QN^2\log N\) uniformly for every \(Q\) in \((A,R_0]\), not only at \(Q=R_0\) -- this uniformity in \(Q\), which (BDH) supplies "for free" because it holds at every \(Q\le t\), is exactly the ingredient partial summation needs and a bound only at the single endpoint \(Q=R_0\) would not supply.
Partial summation against the weight. \(w(q):=\mu(q)^2/\phi(q)^2\) satisfies \(w(q)\le C(\log\log q)^2/q^2\) for \(q\ge3\) (from the classical \(\phi(q)\gg q/\log\log q\)), so \(w(q)\le W(q):=C(\log\log q)^2/q^2\), a majorant that is eventually non-increasing. Since \(f(q)\ge0\), \[ S(N,B)\ \le\ \sum_{q=A+1}^{R_0}W(q)f(q) =W(R_0)F(R_0)+\sum_{q=A+1}^{R_0-1}\big(W(q)-W(q+1)\big)F(q) \] (Abel summation; \(F(A)=0\)). The boundary term is negligible: \(W(R_0)F(R_0)\ll(1/R_0^2)(\log\log R_0)^2\cdot R_0N^2\log N= N^2\log N(\log\log R_0)^2/R_0\), and \(R_0\asymp\sqrt N/(3L)\to\infty\), so this term is \(o(N^2)\) trivially. For the sum, \(W(q)-W(q+1)\ge0\) eventually and \(F(q)\ll qN^2\log N\), so \[ \sum_{q=A+1}^{R_0-1}(W(q)-W(q+1))F(q)\ \ll\ N^2\log N\sum_{q=A+1}^{R_0-1}q\big(W(q)-W(q+1)\big). \] A second partial summation (or direct comparison with \(\int_A^{R_0}x\,d(-W(x))\), integrable since \(xW(x)\sim(\log\log x)^2/x\)) gives \(\sum_{q=A+1}^{R_0-1}q(W(q)-W(q+1))\ll\sum_{q=A+1}^{R_0}W(q)\ll (\log\log A)^2/A\). Combining, \[ S(N,B)\ \ll\ N^2\log N\cdot\frac{(\log\log A)^2}{A} \ +\ o(N^2),\qquad A=\lfloor L^B\rfloor. \] At \(A\asymp(\log N)^B\), \(\log N/A\asymp(\log N)^{1-B}\), so \[ \boxed{\ S(N,B)\ \ll_B\ N^2(\log N)^{\max(1-B,\,0)}(\log\log N)^{O(1)}.\ } \] This is \(O(N^2)\) for \(B=1\) and \(o(N^2)\) for \(B>1\) -- and, since any fixed power of \(\log N\) (or \(\log\log N\)) is \(O_\epsilon(N^\epsilon)\) for every \(\epsilon>0\), this proves \(S(N,B)=O_\epsilon(N^{2+\epsilon})\) for every fixed \(B>0\), exactly the statement asked about, at the full stated range \(q\le R_0\), directly from classical (BDH) alone.
What this is and is not. This is an elementary partial-summation argument built entirely from ingredients this hunt already has on record (classical BDH, cited not re-derived; the same per-\(t\) crude bridging step RANK3_MEAN_VALUE_TOOLS.md §5 already uses). It is not a citation of a "\(t\)-integrated BDH" theorem, because Section 2 does not find one; it shows that, for this specific weight, none is needed. It is this document's own derivation and has not been checked by a second, independent party; Section 4 checks it numerically as far as reachable scale allows.
4. Numerical check: exact but too far pre-asymptotic to confirm or refute the rate
polyrange_weighted_bdh_check.py computes \(S(N,B)\) (discrete-\(t\) form) exactly -- true von Mangoldt enumeration, closed-form per-residue-class partial sums (no sampling, no \(O(N)\)-per-\(t\) loop; validated against a brute-force \(O(N)\) per-\(t\) computation on small cases to floating-point precision, see the script's own development notes) -- for \(N\) from \(3\times10^5\) to \(2.4\times10^6\), \(B=1\) (results in results_polyrange_weighted_bdh_check.json; \(B=2\) needs \(R_0>\lfloor L^2\rfloor\), which needs \(N\) far beyond what this session's compute budget affords -- \(R_0\) only reaches \(14\) to \(35\) across the \(N\) range tested here, and \(B=2\) is reported empty at every \(N\) tried, not silently skipped).
At these \(N\), \(S(N,1)/N^2\) rises from \(0.0045\) to \(0.018\) across an 8-fold range of \(N\) -- not the near-flat behavior Section 3's \((\log\log N)^{O(1)}\) bound would suggest at \(B=1\) (where the bound is \(O(N^2)\) with only log-log-power fluctuation, i.e. \(S/N^2\) should be nearly constant, not visibly rising). This is not read as evidence against Section 3's bound: \(R_0\) itself is only \(14\) to \(35\) at these \(N\) -- rank3_bdh_probe.py's own experiment 1 already found and flagged that (BDH)'s measured \(Q\)-dependence at comparably small \(Q\) (\(10\) to \(500\), with \(N\) fixed) rises noticeably faster than the \(Q\log Q\) asymptotic shape, attributing this to a pre-asymptotic regime where lower-order terms have not yet become negligible relative to the leading term -- and \(R_0\) here is smaller still, and also growing with \(N\) (not held fixed), so the two data points compare genuinely different, non-overlapping \(q\)-windows, not the same window at different \(N\). This numerical check cannot, at reachable \(N\), distinguish "obeys Section 3's bound with a pre-asymptotic transient" from "grows faster than Section 3 claims"; it is reported as measured, not fit to either reading, exactly the caution rank3_bdh_probe.py and RANK3_MEAN_VALUE_TOOLS.md §3 already apply to their own small-\(Q\) measurements. What the same script does confirm cleanly: at every \(N,B\) checked, \(S(N,B)\) computed with the weight \(\mu(q)^2\) alone (no \(1/\phi(q)^2\)) -- i.e. RANK3_ROUTE_D.md (D11)'s actually-proven weight, Section 5 below -- is one to four orders of magnitude larger than \(S(N,B)\) with the \(\mu(q)^2/\phi(q)^2\) weight at the same \(N,q\)-range (e.g. at \(N=2.4\times10^6\): \(1.05\times10^{11}\) versus \(2.68\times10^{13}\)), consistent with, though not a proof of, Section 3's claim that the extra \(1/\phi(q)^2\) factor is doing real, substantial work.
5. The caveat this document inherits from RANK3_ROUTE_D.md: this weight is not the one proven needed
RANK3_ROUTE_D.md §5, working from the same assignment's quoted \(\mu(q)^2/\phi(q)^2\) guess, derives (D7)-(D11) exactly (Parseval, character orthogonality, Cauchy-Schwarz -- no approximation) and finds the guess is not what its own derivation establishes: the weight (D11) actually proves, without assuming any cancellation in the cross term \(\Sigma_{\rm cross}(q)\), is \(2\mu(q)^2\) -- \(O(1)\), with no \(1/\phi(q)^2\) at all. Reaching \(\mu(q)^2/\phi(q)^2\) would require \(\Sigma_{\rm cross}(q)\) to cancel the diagonal term down by a full extra factor of \(\phi(q)^2\) beyond assuming it is merely negligible -- a claim RANK3_ROUTE_D.md explicitly states is undetermined by anything in UPPER_BOUND.md or RESULTS.md, and does not resolve.
Consequently: confirming \(S(N,B)=O_\epsilon(N^{2+\epsilon})\) (Section 3 above) does not, by itself, resolve RANK3_ROUTE_D.md (D11)'s bound on \(U_{(q)}\), because (D11)'s proven weight is \(\mu(q)^2\), not \(\mu(q)^2/\phi(q)^2\). Plugging Section 3's method in with the proven weight instead (\(w(q)=O(1)\), no decay) reproduces exactly RANK3_MEAN_VALUE_TOOLS.md's own (E1)-(E2): the Abel-summation boundary term \(w(R_0)F(R_0)\asymp R_0N^2\log N\asymp N^{5/2}\) now dominates (since \(w\) no longer decays, there is nothing for partial summation to save), and Section 3's bound degrades exactly to RANK3_MEAN_VALUE_TOOLS.md §5's already-recorded \(N^{5/2}\) order -- matching, not beating, CHHL's own GRH-conditional benchmark, the same conclusion that document already reached by a different (dyadic block, unweighted) route. This document adds no improvement to that conclusion; it only pins down, precisely, that the \(\phi(q)^{-2}\) weight is exactly the ingredient separating a bound that closes (Section 3, but for a weight not known to be the right one) from a bound that does not (RANK3_MEAN_VALUE_TOOLS.md's, for the weight that is proven).
6. Verdict
- Literature search (Section 2): no theorem of the exact "\(t\)-integrated BDH" shape is found, under any name checked, working from this hunt's own cited sources and general knowledge of the strands the task names (Bombieri-Vinogradov, Barban-Davenport-Halberstam and its Montgomery/Hooley sharp form, Motohashi, Friedlander-Iwaniec, Fouvry, Bombieri-Friedlander-Iwaniec, variance-of-primes and Barban-Vehov weighted sieve literature) -- with no live network access to confirm or overturn this (
WebSearch/WebFetchboth denied, same wall asRANK3_BDH_VERIFY.md). - But the specific statement asked about is true anyway (Section 3): \(S(N,B)=O_\epsilon(N^{2+\epsilon})\) for every fixed \(B>0\), at the full range \(q\le R_0\), follows from classical (BDH) alone via elementary partial summation against the \(\mu(q)^2/\phi(q)^2\) weight -- no additional theorem, named or not, is needed for this weighted statement. This is this document's own derivation, not a citation.
- Numerically (Section 4): exact at every point checked, but the reachable \(q\)-range (\(R_0\) only \(14\)-\(35\)) is too small and too far pre-asymptotic to empirically confirm or refute the claimed rate, matching the same caution this hunt's own BDH probe already applies to itself at comparably small \(Q\).
- The catch (Section 5, inherited from
RANK3_ROUTE_D.md): the \(\phi(q)^{-2}\) weight that makes Section 3's argument close is not the weightRANK3_ROUTE_D.md(D11) proves is actually needed for \(U_{(q)}\) -- that weight is the larger \(\mu(q)^2\) (no \(\phi(q)\) division), for which the identical method only reproducesRANK3_MEAN_VALUE_TOOLS.md's already-recorded \(N^{5/2}\), not \(N^{2+\epsilon}\). So this document resolves the literature-search question it was assigned, but that resolution does not, on its own, unlockRANK3_ROUTE_D.md's \(U_{(q)}\) bound; what would still be needed for that is a proof (not an assumption) thatRANK3_ROUTE_D.md(D7)'s cross term \(\Sigma_{\rm cross}(q)\) genuinely cancels the diagonal term down toward the \(\phi(q)^{-2}\)-weighted size, which neither that document nor this one supplies.
This document is a literature search (Section 2, negative, network-access limited), one elementary partial-summation derivation built from a classical theorem this hunt already cites (Section 3), a numerical measurement reported with its own pre-asymptotic limitation stated plainly (Section 4), and a reconciliation with RANK3_ROUTE_D.md's independent, exact finding about which weight is actually proven (Section 5); it assumes and establishes nothing about zeros of \(L\)-functions or the Riemann Hypothesis.