/teal-sea
teal-sea / zeta-labstate of record · compiled 28 Sep 2026 · revision e4945c4 · source

Library · hunts/prime_pair_error/RANK3_INTEGRATED_BDH.md

An integrated-in-\(t\) Barban-Davenport-Halberstam theorem at order

2,464 words · 283 lines · source

\(QN^{1+\epsilon}\): not in the cited literature, and not true either

This document answers the question RANK3_MEAN_VALUE_TOOLS.md Section 5 and RANK3_BDH_VERIFY.md leave open: does an "integrated" Barban-Davenport- Halberstam-type theorem exist, or can one be proved, bounding \[ \Sigma(N,Q):=\sum_{q\le Q}\ \sum_{\substack{a\bmod q\\(a,q)=1}}\ \sum_{t=1}^N \big(\psi(t;q,a)-t/\phi(q)\big)^2 \] at order \(QN^{1+\epsilon}\) for every fixed \(\epsilon>0\), uniformly for \(Q\) up to a positive power of \(N\) such as \(R_0=\lfloor\sqrt N/3\rfloor/L\) (\(L=\log N\))? The answer is no on both counts asked for: no such theorem is stated in the standard sources this document can check, and, more importantly, \(\Sigma(N,Q)\) itself is not that small -- it is measured directly below, from the true von Mangoldt function with no sampling, to scale like \(N^{2}\) (up to a log factor), not \(N^{1+\epsilon}\) for any \(\epsilon\) as large as \(0.3\), at every \(Q\) tested. This is not merely "no theorem of this shape is known"; it is "the quantity the theorem would have to bound does not have the claimed order," which is a stronger and more useful negative finding for RANK3_ROUTE_D.md's identity (D11) than a literature gap alone would be.

Notation is RANK3_ROUTE_D.md's and RANK3_MEAN_VALUE_TOOLS.md's: \(\Delta(t;q,a)=\psi(t;q,a)-t/\phi(q)\), \(T(q,b)=\sum_{t=1}^N\Delta(t;q,b)^2 +\sum_{t=1}^{N-1}[\Delta(N;q,b)-\Delta(t;q,b)]^2\) (RANK3_ROUTE_D.md (D5)). \(\Sigma(N,Q)=\sum_{q\le Q}\sum_b^*\sum_{t=1}^N\Delta(t;q,b)^2\) is exactly the first summand of \(\sum_{q\le Q}\sum_b^*T(q,b)\); the second summand is controlled the same way by \((x-y)^2\le2x^2+2y^2\) (RANK3_MEAN_VALUE_TOOLS.md's (E1)) and does not change any order discussed here, so this document works with \(\Sigma(N,Q)\) itself throughout, exactly as the task assigning this document phrases the target quantity.

1. Sourcing: no live network access, same wall as RANK3_BDH_VERIFY.md

This attempt tried WebSearch three times, on the three literature threads the assignment names (an integrated/mean-square-in-\(t\) Barban-Davenport- Halberstam statement; Motohashi's induction principle for generalizations of Bombieri's prime number theorem; the Barban-Vehov weighted sieve and large-sieve/zero-density theorems for Dirichlet \(L\)-functions averaged jointly over conductor and height). All three calls returned a permission error with no interactive user available to grant it -- the identical wall RANK3_BDH_VERIFY.md records ("WebSearch and WebFetch calls both returned a permission error"). No local copy of Motohashi's Sieve Methods and Prime Number Theory, Montgomery's LNM 227, Montgomery and Vaughan's Multiplicative Number Theory I, or Davenport's Multiplicative Number Theory exists in this worktree. Section 2 below is therefore sourced from the standard, checkable content of these works as commonly stated -- the same standing RANK3_BDH_VERIFY.md's citations have -- not from a live fetch; a later attempt with working network access should confirm the specific claims below directly against the primary sources and correct anything a live check contradicts. Section 3, by contrast, is a direct numerical measurement from this repository's own von Mangoldt machinery and needs no external source at all.

2. The three named candidates, examined structurally

Barban-Vehov weighted sieve. This is a linear-sieve device (Barban and Vehov 1968; see Motohashi's Sieve Methods and Prime Number Theory, or Halberstam and Richert, Sieve Methods, for the standard weighted-sieve treatment; it is also the technical predecessor to results like Chen's theorem on primes and almost-primes). Its role is to attach smooth weights to a sifting problem so that a linear combination of sifting functions majorizes or minorizes a target counting function, trading exactness for tractability in a sieve, not a mean-value-theorem, sense. It is a tool for bounding counts of primes (or almost-primes) satisfying multiplicative conditions, and its output is a one-parameter (in the sifting level) inequality, not a statement about \(\psi(t;q,a)-t/\phi(q)\) integrated over an interval of \(t\)-values at all. Nothing in the standard presentations of this method produces, as a byproduct or a corollary, a mean-square bound summed over \(t\); it addresses a different question (counting elements of a sifted set) with a different structure (a single sifting level, not two independent parameters \(q\) and \(t\)). This document finds no route from Barban-Vehov to \(\Sigma(N,Q)\).

Motohashi's induction principle. Motohashi (1976, and the treatment in his Sieve Methods and Prime Number Theory) proves an induction principle that extends Bombieri-Vinogradov-type theorems to a wider class of sequences and weightings than the classical prime-counting case -- it is a method for generalizing which sequence a Bombieri-Vinogradov-shaped theorem applies to (replacing \(\Lambda(n)\) with more general multiplicative or sieve-theoretic weights), by an induction on the number of prime factors removed. What it does not change is the shape of the conclusion: like (BV) itself (RANK3_MEAN_VALUE_TOOLS.md Section 1), the output is a bound on \(\sum_q\max_y\max_a|\Delta|\) (or a mean-square analogue at a single fixed endpoint), for the wider class of sequences, not a statement integrated over an interval of endpoints \(t\le N\). Applying it in place of (BV) would reproduce RANK3_MEAN_VALUE_TOOLS.md Section 2's finding almost verbatim -- an arbitrary saving in the sieve/log parameter at a single endpoint, generalized to more sequences, never a saving across the \(t\)-sum -- so this candidate does not change RANK3_MEAN_VALUE_TOOLS.md's or RANK3_BDH_VERIFY.md's conclusion, it only widens the class of sequences the same conclusion would apply to.

Large-sieve/zero-density theorems jointly averaged over conductor and height. These exist (work descending from Bombieri's and Montgomery's large-sieve density theorems for zeros of \(L(s,\chi)\), further refined by Jutila, Huxley and others, bounding \(\sum_{q\le Q}\sum_\chi N(\sigma,T,\chi)\) jointly in the conductor \(q\) and the zero-counting height \(T\)) and are exactly the mechanism behind (BV) and (BDH) themselves (RANK3_MEAN_VALUE_TOOLS.md Section 1's "dual, via Gauss sums" remark; density theorems are one standard route to zero-free-region-flavored improvements of the large sieve). The "height" \(T\) in these theorems is the height of a zero of \(L(s,\chi)\) on the critical strip, entering through a truncated explicit formula at a single fixed \(x\) (here, \(x=N\)); it is not the same parameter as "the endpoint \(t\) of a partial sum \(\psi(t;q,a)\)," even though both are sometimes loosely called an averaging "over height." Averaging jointly over conductor and zero-height sharpens the range and log-saving of a single-endpoint statement (this is exactly how the effective forms of (BV) are proved); it supplies nothing about integrating the resulting discrepancy over many different values of the sum's upper limit \(t\), which is a completely different kind of average (over the argument at which \(\psi\) is evaluated, not over the zeros used to estimate it at one fixed argument). This document finds these theorems to be a plausible source of an improved log-saving on the single-endpoint quantity -- precisely RANK3_BDH_VERIFY.md's H* territory -- and not a source of the \(t\)-sum strengthening at all.

Conclusion of this section. None of the three named candidates supplies a genuinely \(t\)-integrated mean-square theorem. This matches, and gives structural reasons for, RANK3_MEAN_VALUE_TOOLS.md Section 5's and RANK3_BDH_VERIFY.md Section 6's own "not found" verdicts, extended here to two candidates neither prior document examined by name (Barban-Vehov, Motohashi's induction principle).

3. Why no such theorem can hold at the claimed strength: \(\Sigma(N,Q)\)

is measured to be order \(N^2\), not \(N^{1+\epsilon}\)

Section 2 explains why the literature does not supply the target theorem. This section gives the stronger reason why it could not, regardless of what future method is tried: \(\Sigma(N,Q)\) is a sum of nonnegative terms, and it is both heuristically expected, and measured here directly, to grow quadratically in \(N\), not as \(N^{1+\epsilon}\) for any fixed \(\epsilon\).

The heuristic reason, stated precisely. Fix \(q\) and consider \(\Delta(t;q,b)\) as \(t\) ranges over \(1,\dots,N\). Under the square-root- cancellation heuristic that is both the unconditional content of the sharp Montgomery-Hooley form of (BDH) (valid for \(Q\) close to \(t\)) and the GRH-conditional expectation at every \(Q\) (via \(\psi(t,\chi)=O(\sqrt t\log^2(qt))\) for each nonprincipal \(\chi\bmod q\), combined with near-independence across the \(\phi(q)-1\) characters, which gives \(\Delta(t;q,b)=O(\sqrt{t/\phi(q)}\log^2t)\) after dividing by \(\phi(q)\)), \(\Delta(t;q,b)^2\) is genuinely of size \(\asymp t/\phi(q)\) (times a bounded power of \(\log t\)) for typical \(t\), not asymptotically smaller. This is not an artifact of a weak proof method: it is the same order (BDH) itself asserts as an asymptotic equality, not merely an upper bound, in the range where that has been proved (Montgomery 1970, Hooley). Since \(\Delta(t;q,b)^2\ge0\) for every \(t\), no cancellation across different values of \(t\) is possible in \(\sum_{t=1}^N\Delta(t;q,b)^2\) -- unlike \(\sum_t\Delta(t;q,b)\) itself (signed), the squared sum cannot be made smaller than the sum of its parts by cancellation between terms at different \(t\). Consequently, if \(\Delta(t;q,b)^2\) is genuinely \(\asymp t/\phi(q)\) (up to logs) for even a positive proportion of \(t\in[N/2,N]\), then \[ \sum_{t=1}^N\Delta(t;q,b)^2\ \ge\ \sum_{t=N/2}^N\Delta(t;q,b)^2\ \gg\ \frac N2\cdot\frac{N/2}{\phi(q)}\ \asymp\ \frac{N^2}{\phi(q)}, \] forcing \(\Sigma(N,Q)=\sum_{q\le Q}\sum_b^*\sum_t\Delta(t;q,b)^2\) to be \(\gg\sum_{q\le Q}\phi(q)\cdot N^2/\phi(q)\asymp QN^2\) -- a bare power \(N^2\), not \(N^{1+\epsilon}\) for any \(\epsilon<1\). Nothing about GRH changes this: GRH controls the pointwise size of \(\Delta(t;q,b)\) at each individual \(t\), it does not make \(\Delta(t;q,b)\) atypically small at most values of \(t\) up to \(N\) -- if anything, GRH's own square-root size is exactly consistent with, not smaller than, the order just derived. Reaching \(\Sigma(N,Q)=O(QN^{1+\epsilon})\) would require \(\Delta(t;q,b)^2\) to be \(o(t^{1-\delta}/\phi(q))\) for almost every \(t\le N\), for some fixed \(\delta>0\) -- a claim substantially stronger than square-root cancellation (which gives \(\Delta(t;q,b)^2\asymp t/\phi(q)\), not smaller), and one this document finds no support for anywhere, conditionally or otherwise.

Direct numerical measurement. rank3_integrated_bdh_probe.py (results in results_rank3_integrated_bdh_probe.json) computes \(\Sigma(N,Q)\) exactly (true von Mangoldt function by prime-power enumeration, exact cumulative sums per residue class, no sampling) for \(Q\in\{5,10,20,30\}\) and \(N\) from \(10^4\) to \(3.2\times10^5\) (a 32-fold range), reusing rank3_bdh_probe.py's already-cross-checked von_mangoldt and sympy's totient for \(\phi(q)\). Two findings:

This is measured only at \(Q\le30\), well below \(R_0\asymp\sqrt N/(3L)\); it does not, on its own, rule out some qualitatively different behavior emerging only at \(Q\) polynomially large in \(N\). But nothing in this measurement, or in RANK3_MEAN_VALUE_TOOLS.md Section 3's separate finding that \(D(N,Q)/(QN)\) at fixed \(N=150000\) rises faster than \(\log Q\) as \(Q\) grows from \(10\) to \(500\), suggests growing \(Q\) helps -- if anything both point the other way. This document reports the measured range honestly and does not extrapolate a proof for all \(Q\le R_0\) from it; the combination of the heuristic argument (Q-independent: it holds for each fixed \(q\) individually, before any sum over \(q\)) and this numerical confirmation is offered as strong evidence, not a proof, that no \(Q\) range changes the conclusion.

4. What this means for RANK3_ROUTE_D.md's identity, and what would

actually be needed

RANK3_ROUTE_D.md's (D11) needs \(\sum_{q\le R_0}\sum_b^*T(q,b)\) small; since \(\Sigma(N,Q)\) is (Section 3's first summand of) exactly this quantity, and is genuinely order \(QN^2\)-ish rather than \(QN^{1+\epsilon}\), no integrated mean-value theorem of the shape asked for exists to plug in, and none should be expected to be found by further search, because the target quantity itself does not have that order -- the obstruction is not a gap in current technique but the true size of the object. This sharpens RANK3_MEAN_VALUE_TOOLS.md Section 5's and RANK3_BDH_VERIFY.md's "not found" into "not true," at least at the \(Q\) range this document could measure, under the heuristic that is consistent with both the proved sharp (BDH) asymptotic and with GRH.

What would actually be needed instead, stated as precisely as this document can: not a smaller bound on \(\Sigma(N,Q)\) itself, but an argument that never needs to bound the full nonnegative sum \(\sum_b^*T(q,b)\) in the first place -- i.e. a route to \(U_{(q)}\) that exploits cancellation between different residues \(b\) (RANK3_ROUTE_D.md's \(\Sigma_{\rm cross}(q)\), (D7)) rather than a mean-value bound on the diagonal \(\sum_b^*T(q,b)\) alone. This is exactly the open question RANK3_ROUTE_D.md Section 3 and Section 5 already name and do not resolve -- whether \(\Sigma_{\rm cross}(q)\) is close to \(0\) or close to its Cauchy-Schwarz ceiling -- and it is a genuinely different kind of question from the one this document answers: it asks about cancellation across residues at fixed \(t\), not about the size of \(\Delta(t;q,b)^2\) summed across \(t\) at fixed residue, which is what this document shows cannot be beaten down to \(N^{1+\epsilon}\) order. A route through \(\Sigma_{\rm cross}(q)\), if it existed, would sidestep this document's obstruction entirely rather than contradict it, since it would never form \(\sum_b^*T(q,b)\) as an intermediate quantity to begin with.

5. Verdict

This document is a citation-structure analysis (Section 2, sourced as Section 1 describes, without live network access), a heuristic argument from elementary nonnegativity plus the classical (BDH) asymptotic and the GRH square-root-cancellation heuristic (Section 3), and a direct, exact numerical measurement from this repository's own von Mangoldt machinery (Section 3, results_rank3_integrated_bdh_probe.json); it assumes and establishes nothing about zeros of \(L\)-functions or the Riemann Hypothesis beyond citing GRH's standard conjectural consequence for \(\psi(t,\chi)\) as one of two independent heuristics compared against measured data, not as an assumption relied upon for the verdict above.