# Möbius Bombieri–Vinogradov: published inputs and a self-contained derivation Internal note, 2026-09-06. Provenance repair only. **Nothing about the twin-prime estimate moves here.** The estimate below is classical and was already used in [shifted-prime-decomposition.md §2](shifted-prime-decomposition.md); the defect recorded in [chain-review-0906.md](chain-review-0906.md) was that its only cited source was an exercise sheet. Finite checks of the identity and of the support claims are in [mobius-bv-validation.js](mobius-bv-validation.js). **Local readiness review:** review 21 rechecked the full local derivation and the author-preliminary source statements K1--K4. In the small-y split of §3, apply K1 only for y>=T/log^D T, so log y is comparable to log T and the excluded-prime condition holds with y itself; below this threshold use the trivial bound. Source theorems remain imports. ## 1. The statement For every fixed A>0 there is B=B(A)>0 such that, uniformly for T≥3 and Q≤T^(1/2)/(log T)^B, $$ \sum_{q\le Q}\ \max_{(a,q)=1}\ \max_{y\le T} \Bigl|\sum_{n\le y,\ n\equiv a\ (q)}\mu(n)\Bigr| \ \ll_A\ \frac{T}{(\log T)^{A}}. \tag{M} $$ (M) is the statement labelled (4) in `shifted-prime-decomposition.md`, with the maximum over y≤T added. The added maximum is free in the derivation below and makes that note's mesh argument for interval endpoints unnecessary, since a sum over a subinterval is a difference of two terms of the kind bounded in (M). The mesh argument there is not wrong; it is simply no longer needed. Calibration: **derived here from published theorems, not itself published, and not reviewed by anyone outside this repository.** ## 2. Is there a published theorem statement? No, in five channels Searched in the owning convention recorded in [SEARCH-CONVENTIONS.md §1](SEARCH-CONVENTIONS.md) ("Möbius function in arithmetic progressions", "Bombieri–Vinogradov", "Vaughan identity"), 2026-09-06. | source | what it actually contains | status | |---|---|---| | Granville–Shao, *Bombieri–Vinogradov for multiplicative functions, and beyond the x^(1/2)-barrier*, Adv. Math. **350** (2019) 304–358 | p. 2, read byte-level in the arXiv PDF ([1703.06865](https://arxiv.org/pdf/1703.06865)): "The analogous result is known to hold when f = µ, the Mobius function, and when f is the characteristic function for the y-smooth numbers [16, 23]". References [16] and [23] are Fouvry–Tenenbaum and Harper, both **smooth-number** papers; no locator is attached to the µ half of the sentence. The same paragraph points at "chapter 28 of [8]" (Davenport) and "Theorems 9.16, 9.17, and 9.18 of [18]" (Friedlander–Iwaniec, *Opera de Cribro*) for proofs of **the original** theorem, that is for primes. | published **assertion** that µ is known; **no proof locator**. Read in the arXiv PDF, not in the journal version | | Koukoulopoulos, *The Distribution of Prime Numbers*, AMS GSM **203** (2019) | Full text of the author's preliminary version searched. Chapter 26 proves Bombieri–Vinogradov for primes; its Exercises 26.1–26.5 do not include µ. What the book does contain for µ is listed in §3 below. | no µ Bombieri–Vinogradov statement; **byte-level negative in a source that has every input** | | Iwaniec–Kowalski, *Analytic Number Theory*, AMS Colloq. **53** (2004) | Chapter 17 "Primes in Arithmetic Progressions", §17.2 "Bilinear forms in arithmetic progressions" (p. 421), §17.3 "Proof of the Bombieri–Vinogradov Theorem" (p. 423): section titles and page numbers confirmed byte-level in the AMS-hosted `coll053-endmatter.pdf`. The theorem text of §17.2, including the statement usually cited as Theorem 17.4, **could not be opened**. | **NOT REACHED**; a general bilinear theorem there would be the natural published carrier | | Friedlander–Iwaniec, *Opera de Cribro*, AMS Colloq. **57** (2010), Theorems 9.16–9.18 | Cited by Granville–Shao as an elegant proof of the prime case. Text closed access on every channel tried. | **NOT REACHED** | | Davenport, *Multiplicative Number Theory*, ch. 28 | Chapter title "Bombieri's Theorem" confirmed at the publisher's chapter listing; text behind a login. Cited by Granville–Shao for the prime case. | **NOT REACHED**; no reason to expect a µ statement | | Tao, 254A Notes 3, Theorem 16 | General bilinear Bombieri–Vinogradov, statement fetched and read (§3 below). µ appears only in Exercise 14, which is the Barban–Davenport–Halberstam L² statement, not (M). | blog, not published | | Le Boudec, EPFL Exercise Sheet III, Exercise 4 | States (M) without the maximum over y. | teaching material; the defect being repaired | So: **no published theorem statement of (M) was located**, searching the owning convention of [SEARCH-CONVENTIONS.md §1](SEARCH-CONVENTIONS.md). That is a negative about five channels searched on one day, not a proof that none exists; the two most likely carriers (Iwaniec–Kowalski §17.2 and *Opera de Cribro* §9) were not opened, and either could contain a general theorem that covers µ once a Vaughan decomposition is supplied. What would settle it: page access to those two. ## 3. The published inputs used below All four are numbered results in Koukoulopoulos, *The Distribution of Prime Numbers*, AMS Graduate Studies in Mathematics **203** (2019). Read in the [author's preliminary version](https://dms.umontreal.ca/~koukoulo/documents/publications/primes.pdf), posted with the publisher's permission. **Caveat: numbering is from that preliminary version and was not checked against the printed book.** Throughout, $$ \Delta_f(y;q,a)=\sum_{n\le y,\ n\equiv a\ (q)}f(n) -\frac1{\varphi(q)}\sum_{n\le y,\ (n,q)=1}f(n). $$ **K1 (Corollary 13.4).** Fix A,C≥1 and ε∈(0,1/2]. For x≥2 and q,m∈N with q≤(log x)^C and ω(m)≤exp{(log x)^(1−ε)}, $$ \sum_{n\le x,\ (n,m)=1,\ n\equiv a\ (q)}\mu(n)\ \ll_{\varepsilon,A,C}\ x/(\log x)^{A}. $$ This is the Siegel–Walfisz input for µ. Three features are used below and all three are in the statement: a is arbitrary (the proof splits off d=(a,q)), the coprimality condition to m is included, and the modulus range is a fixed power of the logarithm. The constant is ineffective, since the proof runs through the book's Theorem 12.1, which uses Siegel's theorem (Theorem 12.10). **K2 (Theorem 26.2).** Let v≥0 and let f be supported on [1,y]. For x≥2, q∈N, a∈(Z/qZ)*, |Δ_{f∗log^v}(x;q,a)| ≤ 2(log x)^v Σ_{k≤y}|f(k)|. Only v=0 is used, so log^v is the constant function 1 and the bound is 2Σ_{k≤y}|f(k)|, uniform in x and in the reduced class a. **K3 (Theorem 26.6).** Let f, g be supported on [1,M], [1,N]. For x,P≥1, $$ \sum^{*}_{q\le P,\ \chi\ (q)}\frac q{\varphi(q)} \max_{y\le x}\Bigl|\sum_{n\le y}(f*g)(n)\chi(n)\Bigr| \ll\bigl(\sqrt{MN}+\sqrt MP+\sqrt NP+P^2\bigr)(\log x)\,\|f\|_2\|g\|_2, $$ the star restricting to primitive χ. The maximum over y is inside, obtained in the book by Perron inversion; this is where (M)'s maximum over y comes from. **K4 (equation (26.3)).** For any arithmetic function f, q∈N and a∈(Z/qZ)*, Δ_f(y;q,a) = φ(q)^(−1) Σ_{χ (q), χ≠χ0} χ̄(a) Σ_{n≤y} f(n)χ(n). An alternative to K2+K3+K4 is [Tao, 254A Notes 3, Theorem 16](https://terrytao.wordpress.com/2015/01/10/254a-notes-3-the-large-sieve-and-the-bombieri-vinogradov-theorem/), read 2026-09-06: for α on [1,M], β on [1,N], MN≪x, with Σ|α(m)|²≪M log^O(1)x, Σ|β(n)|²≪N log^O(1)x and the Siegel–Walfisz property Δ(β·1_{(·,s)=1};a(r)) ≪_A N log^(−A)x for all A, all primitive a(r) and all 1≤s≤x, one has Σ_{q≤Q} sup_a |Δ(α∗β;a(q))| ≪_A x log^(−A)x for Q≤x^(1/2)log^(−B)x and M,N≥log^B x. That theorem is stated for a single bilinear pair and is a blog source; K2–K4 are preferred because they are in a published book and because the Type I piece is handled without a Siegel–Walfisz hypothesis on it. ## 4. Vaughan's identity for µ For any U,V≥1, with µ_{≤U}(n)=µ(n)1_{n≤U} and µ_{>U}=µ−µ_{≤U}, $$ \mu=\mu_{>U}*\mu_{>V}*1-\mu_{\le U}*\mu_{\le V}*1+\mu_{\le U}+\mu_{\le V}. \tag{V} $$ Proof: expand µ_{>U}∗µ_{>V}∗1 = (µ−µ_{≤U})∗(µ−µ_{≤V})∗1 and use µ∗1=δ three times, giving µ − µ_{≤V} − µ_{≤U} + µ_{≤U}∗µ_{≤V}∗1. Rearranging gives (V). No range restriction on n is needed, in contrast with Vaughan's identity for Λ. (V) is Exercise 23.4(a) of the same book; the two-line proof above makes the exercise status irrelevant. Because 1_{(n,r)=1} is multiplicative, (V) survives multiplication by it with the restriction applied to each factor separately. Measured, not proven: [mobius-bv-validation.js](mobius-bv-validation.js) checks (V) pointwise at every n≤10^5 for eight (U,V,r), three of them with r>1, and checks the support and size claims used in §5 and the exactness of §5's dyadic split. Finite checks say nothing about the asymptotics. ## 5. The derivation Fix A≥1, write L=log T, and take U=V=T^(1/5). Split (V) as µ = µ″ − µ′ + µ_{≤U} + µ_{≤V} with µ′=(µ_{≤U}∗µ_{≤V})∗1 and µ″=µ_{>U}∗(µ_{>V}∗1). Set f=µ_{≤U}∗µ_{≤V}, supported on [1,UV] with |f|≤τ, and g=µ_{>V}∗1, supported on (V,∞) with |g|≤τ. **(a) Type I.** By K2 with v=0, for all y, q and a∈(Z/qZ)*, |Δ_{µ′}(y;q,a)| ≤ 2Σ_{k≤UV}|f(k)| ≤ 2Σ_{k≤UV}τ(k) = 2Σ_{d≤UV}⌊UV/d⌋ ≤ 2UV(log(UV)+1) ≪ T^(2/5)L. Trivially |Δ_{µ_{≤U}}| ≤ U/q+1+U/φ(q). Hence $$ \sum_{q\le Q}\max_{(a,q)=1}\max_{y\le T} \bigl|\Delta_{\mu'}+\Delta_{\mu_{\le U}}+\Delta_{\mu_{\le V}}\bigr| \ \ll\ QT^{2/5}L+QU\ \ll\ T^{9/10}L . $$ **(b) Character reduction of the Type II part.** By K4, and writing each non-principal χ mod q through the primitive ξ mod d inducing it, with d|q, d>1, q=dr and χ(n)=ξ(n)1_{(n,r)=1}, and using φ(q)≥φ(d)φ(r), $$ \sum_{q\le Q}\max_{(a,q)=1}\max_{y\le T}|\Delta_{\mu''}(y;q,a)| \le\sum_{r\le Q}\frac1{\varphi(r)}\sum_{1L^C.** For n≤T, µ″(n)1_{(n,r)=1} = Σ_{U<2^j≤2T/V} (α_j∗β_j)(n) with α_j(k)=µ(k)1_{k>U}1_{2^(j−1)U and l>V, hence k≤T/V. Then ‖α_j‖₂ ≤ 2^(j/2) and ‖β_j‖₂ ≪ (T/2^j)^(1/2)L^(3/2) by Σ_{l≤z}τ(l)²≪z log³z, so ‖α_j‖₂‖β_j‖₂ ≪ T^(1/2)L^(3/2). K3 with M=2^j, N=T/2^(j−1) and any P≥1 gives $$ \sum^{*}_{q\le P,\ \chi\ (q)}\frac q{\varphi(q)}\max_{y\le T} \Bigl|\sum_{n\le y}(\alpha_j*\beta_j)(n)\chi(n)\Bigr| \ll\bigl(T+2^{j/2}T^{1/2}P+T2^{-j/2}P+P^2T^{1/2}\bigr)L^{7/2}, $$ and summing over the ≪L admissible j, with 2^j>U and 2^j≤2T/V, $$ \sum^{*}_{q\le P,\ \chi\ (q)}\frac q{\varphi(q)}\max_{y\le T} \Bigl|\sum_{n\le y}\mu''(n)\chi(n)1_{(n,r)=1}\Bigr| \ll\Bigl(T+\frac{TP}{\sqrt U}+\frac{TP}{\sqrt V}+P^2\sqrt T\Bigr)L^{9/2}. \tag{5.2} $$ Splitting the conductors into dyadic blocks 2^i and applying (5.2) with P=2^i, $$ \sum_{L^CL^C}2^(−i)≪L^(−C), Σ_{2^i≤2Q/r}1≪L and Σ_{2^i≤2Q/r}2^i≪Q/r. Summing over r≤Q with weight 1/φ(r), so that Σ_r 1/φ(r)≪L and Σ_r 1/(rφ(r))≪1, the large-conductor part of (5.1) is $$ \ll\ \frac{T L^{11/2}}{L^{C}}+T^{9/10}L^{13/2}+Q\sqrt T\,L^{11/2}. \tag{5.3} $$ **(e) Conclusion.** Take C=A+6. The first term of (5.3) is ≪T/L^A, the second is ≪T/L^A for large T, and the third is ≪T/L^A as soon as Q≤T^(1/2)/L^(A+6). With (a) and (c) this gives $$ \sum_{q\le Q}\max_{(a,q)=1}\max_{y\le T}|\Delta_\mu(y;q,a)|\ \ll_A\ T/L^{A}, \qquad Q\le T^{1/2}/(\log T)^{A+6}. \tag{5.4} $$ Finally K1 with modulus 1 and m=q bounds the subtracted term: |Σ_{n≤y,(n,q)=1}µ(n)| ≪_A T/L^(A+1) uniformly in q≤T (trivially for y≤T/L^(A+1)), so Σ_{q≤Q}φ(q)^(−1)·T/L^(A+1) ≪ T/L^A. Adding this to (5.4) gives (M) with **B(A)=A+6**. No attempt was made to optimise B. ## 6. What this does and does not settle - The constant in (M) is **ineffective** for A above an absolute threshold, because K1 runs through Siegel's theorem. Every downstream use in this corpus is qualitative, so nothing currently depends on effectivity. Any future explicit-constant claim must not cite (M). - (M) covers **reduced classes only**. Non-reduced classes are not derived here and are not needed: `shifted-prime-decomposition.md §3` reduces every congruence it meets to a primitive class before applying the estimate. - The level is exactly 1/2 with a logarithmic saving. Nothing here touches the x^(1/2) barrier, and the Granville–Shao paper is about going past it for multiplicative f with a fixed residue class, which is a different statement. - Calibration of the whole note: the mathematics is standard and every step is a named published theorem or a two-line verification, but **the assembly is ours and has had one adversarial pass only, by its author.** The failure mode to look for is a hypothesis of K1 or K3 used outside its range; the ranges are stated inline at each application for that reason. - Two small discrepancies noticed in the source and recorded so they are not rediscovered: the final display of the book's Chapter 26 prints x(log x)^5/min{U,V} where its own Corollary 26.7 gives the square roots √U, √V; the conclusion there is unaffected because that proof takes U=V=e^(√log x). The derivation above uses Theorem 26.6 directly and does not depend on either form.