A few remarks on the Baez-Duarte Criterion

Alexandre Pyvovarov

REVIEW 1 major objections 5 minor 19 references

The paper reduces the Riemann-Hypothesis-equivalent boundedness problem for exponentially damped Möbius approximations to proving that one explicit combination of four bilinear sums stays bounded.

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2026-08-02 06:42 UTC pith:FMBB2SYU

load-bearing objection A careful, honest reduction of the boundedness question to one explicit bilinear criterion; the main unproved hinge is the Euler–Maclaurin cut-term bound in Theorem 3.17. the 1 major comments →

arxiv 2607.12084 v3 pith:FMBB2SYU submitted 2026-07-13 math.NT math.CO

A few remarks on the Baez-Duarte Criterion

Alexandre Pyvovarov This is my paper

classification math.NT math.CO MSC 11M2611M0611L0311N37

keywords Riemann HypothesisNyman-Beurling criterionMobius functioncotangent sumsHilbert space approximationfinite-scale decompositionthird-order truncationEuler-Maclaurin

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies a family of exponentially damped approximations to the constant function built from the Möbius coefficients, and the squared norm F(x) as the damping goes to zero. Boundedness of F as x approaches 1 is the known equivalent of the Riemann Hypothesis. The paper constructs a canonical third-order truncation F_[3] by discarding one remainder term, proves an unconditional growth bound for F_[3], and proves that the discarded remainder splits into a one-variable part (harmless) and a two-variable part E_rho3. For E_rho3 it proves an exact finite-scale identity that reduces boundedness to a single combined estimate: log(1/u) A + C + U + V = O(1), where the four terms are explicit bilinear sums. A sympathetic reader would see this as a meaningful reduction: the entire difficulty of the criterion is now confined to one explicit cancellation condition.

Core claim

The central discovery is an exact finite-scale decomposition of the remainder. For 0<u<=1/2, identity (9.36) writes E_rho3(e^{-u}) as log(1/u) A_rho3(u^{-1}) + C_rho3(u^{-1}) + U_rho3(u) + V_rho3(u), where A and C are finite sums over coprime pairs with n+m <= u^{-1} of the Möbius-weighted remainder times a density factor, U is the corresponding dilation-error sum, and V is the far tail. Corollary 9.38 then states that boundedness of E_rho3 (and hence of F) is equivalent to the combined expression being O(1). Along the way the paper proves numerous exact cancellations: the principal residue character is absent from the edge and bulk kernels, the transposition defect cancels identically when

What carries the argument

The canonical third-order truncation F_[3] defined by setting the remainder rho_3 = 0 in every ordered occurrence. Here rho_3(n,a) = πA(n,a)/n - K4(n,a) is the error left after a fixed-residue third-order expansion of the cotangent sum A(n,a); K4 contains seven explicit pieces including the C0..C3 sums. The finite-scale identity (9.36) and the uniform dilation formula W_{n,m} = κ(nm) log(1/(u(n+m))) + β(nm) + ... carry the argument, because they convert the infinite double series into a finite part plus a controllable error.

Load-bearing premise

The entire truncation rests on the asserted but not displayed bound that the omitted Bernoulli-degree-five cut term and the remainder in the shifted Euler-Maclaurin summation are O_H(n^{-4}); if that bound fails, the coefficients defining the canonical truncation would not be valid.

What would settle it

Compute the combined expression log(1/u) A_rho3(u^{-1}) + C_rho3(u^{-1}) + U_rho3(u) + V_rho3(u) at a sequence u -> 0 using high-precision Möbius sums and the explicit formulas for the remainder; if it does not remain bounded, then F(x) is unbounded and the approximation criterion fails. More narrowly, evaluating A_rho3(X) for increasing X and checking whether log X times A_rho3(X) tends to 0 would settle the sufficient criterion.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

If the four displayed terms in (9.40) cancel, then F(x) is bounded as x -> 1, which by the known criterion is the Riemann Hypothesis.
The canonical truncation F_[3] has at most power-logarithmic growth without using any cancellation from the Möbius function, so all hidden cancellation is concentrated in the remainder.
The one-variable remainder R_{1,3} is C^4 on [0,1] and bounded; the transposition defect cancels exactly, so the only genuinely bilinear obstacle is the remainder E_rho3.
The core and fixed-edge remainder terms are O_H(log^2(e/u)) for each fixed H, and the bulk is O(u^{-1} log^2(e/u)); no bound permits H to grow with u.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the combined criterion (9.40) turns out to hold, it would likely be via cancellation between the logarithm term and C rather than via each term being small, since the absolute-value bound in (8.9) is too weak.
The same finite-scale template could be applied to higher-order truncations: replacing rho_3 with a fourth-order remainder would push the displayed coefficients one power of n higher and might make the density sums absolutely convergent, at the cost of more complicated local constants.
The explicit form of A_rho3(X) suggests it can be tested numerically at large X with high-precision Möbius sums; a slow drift would be a concrete way to look for a counterexample to boundedness.
Because the paper shows the principal character is absent from both edge and bulk remainder kernels, the surviving sums are supported on nonprincipal characters; this may make them accessible to the classical nonvanishing of L(1,χ), though the paper does not deploy that here.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

A careful, honest reduction of the boundedness question to one explicit bilinear criterion; the main unproved hinge is the Euler–Maclaurin cut-term bound in Theorem 3.17.

read the letter

Colleague — this is worth reading if you work on Nyman–Beurling or cotangent sums. The paper does not claim RH. It does something more useful: it defines a canonical third-order truncation of Vasyunin's cotangent sum, proves the truncated model grows at most like u^{-1} log^2(e/u), and reduces boundedness of the full F to a single explicit criterion (9.40). The recombination identities (6.11), (8.8), (9.36) are exact and the derivation is structural — no fitted constants, no circularity beyond importing Vasyunin's formula.

The genuinely new pieces are the canonical truncation deleting rho_3, the uniform squarefree dilation formula (5.13) with explicit kappa and beta, the sector decomposition, and the finite-scale criteria. Theorem 7.30 and 9.21 are real asymptotics with explicit coefficients. The paper is also honest about what remains: section 9.6 states clearly that the logarithmic coefficient on the initial edge is not known to vanish, that H cannot tend to infinity, and that the bulk is only controlled with a power loss.

The soft spot is where the reader's report and the stress-test agree: Theorem 3.17's Euler–Maclaurin step. Equation (3.22) asserts the omitted degree-five cut term and remainder are O_H(n^{-4}) uniformly in 1<=a<=H, but the proof only says the one-sided derivatives are bounded by constants depending on H. That is not enough to see the uniformity in n, and the coefficients C0..C3, hence K4 and the whole truncation, rest on it. The stress-test's concern is legitimate: if that bound fails or has an n^{-4} main term, the edge asymptotics (9.15), (9.22) would need revision. This is not a manufactured flaw — it is a gap in the text at a load-bearing point. That said, the exact algebra of the reduction (6.11) and (9.40) would survive; what would collapse is the advertised remainder size, not the structural decomposition itself.

There is also the usual reliance on Vasyunin's external formula; that is prior work and not a flaw by itself. The citation pattern is appropriate — Bettin–Conrey, Maier–Rassias, Darses–Hillion are all relevant and the paper distinguishes its own contribution.

Bottom line: this deserves a serious referee. The referee should ask for a proof of the Euler–Maclaurin remainder bound in Theorem 3.17, or a weakening with explicit hypotheses. It is a useful step for a subfield, not a breakthrough; I would not bet on (9.40) being easy, but the reduction is a real contribution.

Referee Report

1 major / 5 minor

Summary. The paper studies the exponentially damped Möbius approximants f(u) in the Hilbert space H=L^2([1,∞), dt/t^{-2}) and the function F(x)=||f(u)||_2^2 with x=e^{-u}. Starting from Vasyunin's cotangent-sum formula, the author defines a canonical third-order truncation F_[3] by deleting the unique remainder ρ_3 in the ordered primitive-pair expansion. The main results are: exact identities (6.11), (8.8), (9.35)–(9.36) relating F to F_[3] and the remainder terms; a global unconditional bound |F_[3](e^{-u})| ≪ u^{-1} log^2(e/u) (Theorem 8.14); unconditional sector bounds for the remainder terms (Theorem 9.13); and an exact finite-scale criterion (9.40) which is claimed to be equivalent to boundedness of F. The paper does not prove the Riemann Hypothesis or even boundedness of F; its stated aim is to reduce the Baez-Duarte boundedness problem to an explicit global bilinear cancellation condition.

Significance. If the exact identities are correct, the paper provides a rigorous and remarkably explicit reformulation of the key boundedness problem in the Baez-Duarte/Nyman-Beurling approach. The finite-scale decompositions (8.8) and (9.36) are genuine mathematical statements, not heuristic asymptotic substitutions, and the paper is honest about what remains open. The uniform squarefree dilation estimates and the sector bounds are useful technical contributions. The paper does not overclaim: the unresolved part is isolated in (9.40), and several remarks explicitly warn where an absolute-value estimate is insufficient. The main weakness is that the proof of the central fixed-residue expansion (Theorem 3.17) is compressed at a load-bearing point, and the entry via Vasyunin's formula is quoted rather than stated.

major comments (1)

[Theorem 3.17, Eq. (3.22)] The proof of Theorem 3.17 is the pivot of the paper: the coefficients C_0,...,C_3 define K_4 and hence the canonical truncation F_[3], and the estimate (3.18) is used in (9.15) and Theorem 9.21. However, the assertion that the omitted Bernoulli-degree-five cut term and the Euler–Maclaurin remainder are O_H(n^{-4}) is stated without a displayed derivation. The sentence 'all one-sided derivatives required here are bounded by constants depending only on H' is plausible because the cuts j/a are fixed for a≤H and the endpoint singularities have been subtracted, but the estimate must be written out: one needs the blockwise EM remainder formula and a clear verification that no additional n^{-3} or n^{-4} main term arises from the one-sided jumps, uniformly for 1≤a≤H and n→∞. I do not regard this as an actual error, but it is a load-bearing proof gap in the current text and should be fixed befor

minor comments (5)

[Proposition 2.8] The formula for F(x) uses the same letter n in the product of two infinite series; the dummy indices should be distinct. More importantly, the Vasyunin scalar-product identity is imported as 'according to the expressions proven in [Vas96]'; please state explicitly the exact normalization and formula being used, since all subsequent identities depend on it.
[Throughout] There are many typographical and language errors: 'B´ aez-Duarte' with misplaced accent, 'par', 'beacuse', 'deines', 'fiw', 'aplly', and inconsistent use of 'Mobius' vs 'Möbius'. The paper needs careful copyediting.
[Notation] The paper introduces many symbols (W_{n,m}, Σ_{[3]}, Ψ_{[3]}, A_{[3]}, C_{[3]}, E_{[3]}, T_{[3]}, A_{ρ3}, etc.). A table of notation or a short glossary would substantially improve readability, especially because the same letters are used with different meanings in Sections 2 and 9.
[Remark 7.2] The remark correctly warns that (7.18)–(7.19) cannot yet be inserted as coefficients of log(1/u) in the W-weighted Abel sum. This is an important point and should perhaps be made more prominent, since it explains why the principal-character cancellation does not immediately yield the desired bound.
[Theorem 8.14] The final comparison '−log x ≍ 1−x near 1' should be stated as x↑1; as written it is slightly imprecise for x in (0,1).

Circularity Check

0 steps flagged

No significant circularity: the main results are exact decompositions and unconditional bounds; the only external input is Vasyunin's formula, not the author's own prior work.

full rationale

The paper's central objects are defined explicitly rather than fitted. The canonical truncation F_[3] is defined by the algebraic operation of deleting the displayed remainder rho3 in the exact ordered-pair expansion, and the identities F = F_[3] + E_rho3 - m1 R1,3 (6.11), the finite-scale formula (8.8), and the remainder formula (9.36) are exact consequences of the definitions and Vasyunin's cotangent representation. The boundedness criterion (9.40) is explicitly presented as an unresolved equivalent formulation, not as a derived theorem: Section 9.6 states precisely what is proved and what remains, and Remark 7.2 warns that the Dirichlet-series identities (7.18)-(7.19) cannot yet be inserted as Abel asymptotics. There are no fitted constants and no renamed empirical pattern; the unconditional growth bound (8.15) is proved by crude absolute-value estimates, and the paper does not claim to have resolved RH or boundedness. The only load-bearing cited input is Vasyunin's formula [Vas96], which is prior external work not authored by the present author, and no author-specific uniqueness theorem is invoked. The Euler-Maclaurin remainder assertion in (3.22) is a proof gap candidate (the text asserts the omitted degree-five cut term and remainder are O_H(n^-4) with a brief derivative-size justification), but an asserted estimate is a correctness risk, not circularity; it is not equivalent by construction to the paper's conclusions. Therefore no circular step is identified.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No free parameters are fitted; the auxiliary H is bookkeeping and cancels. The main unproved inputs are classical analytic number theory plus Vasyunin's external scalar-product formula. No invented entities are introduced; F_[3], rho_3, and W are definitions, not postulates.

axioms (6)

standard math Prime number theorem and effective Perron formula (Lemma 1.4)
Used to bound alpha(x)=sum_{n<=x} mu(n)/n and to justify convergence and boundary values of Möbius Dirichlet series (Sections 1.3, 2.3, 6).
standard math Korobov-Vinogradov zero-free region and Mertens-type estimate |M(y)| <= C y exp(-c0 Phi(y))
Used in Lemma 2.12 to prove the exponential approximation ||Q_k - g|| << E_{c3}(k), which underpins Proposition 2.19 and the h_k bounds.
domain assumption Vasyunin's scalar product formula for <gamma_n | gamma_m> in the exact normalization used
Quoted from [Vas96] in Proposition 2.8; the paper says the formula can be 'easily inferred' but does not derive it. A normalization error here would propagate into A(n,m), K4, rho_3, and F_[3].
standard math Euler-Maclaurin summation in periodic-Bernoulli form, including the asserted O_H(n^{-4}) bound for the omitted degree-five cut term
Used in Theorem 3.17 to derive the fixed-residue third-order expansion; the cut-term remainder is asserted rather than fully displayed.
standard math Classical nonvanishing of Dirichlet L-functions at s=1 for nonprincipal characters
Used in Section 7.2 to prove existence of the Dirichlet-Abel densities D_a (equations 7.20-7.21).
standard math Standard squarefree-counting and Dirichlet-series factorization, e.g., sum mu(d)^2/d^s = zeta(s)/zeta(2s) and Euler products for kappa(r)
Used in Theorems 5.10 and 5.12 for the uniform dilation kernel D_r(e^{-y}).

pith-pipeline@v1.3.0-alltime-deepseek · 33151 in / 27587 out tokens · 237006 ms · 2026-08-02T06:42:47.732595+00:00 · methodology

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read the original abstract

We study exponentially damped M\"obius approximants in $\mathscr H=L^2([1,\infty),dt/t^{-2})$. With \[ \gamma_n(t)=\left\lfloor\frac tn\right\rfloor -\frac{\lfloor t\rfloor}{n},\qquad f(u)(t)=\sum_{n\ge1}\mu(n)e^{-nu}\gamma_n(t),\] we compute the relevant scalar products, characterize the M\"obius coefficients as the unique coefficients giving pointwise convergence to the constant function, and prove $\langle1 \mid f(u)\rangle\to1$. Vasyunin's formula expresses $F(e^{-u})=\|f(u)\|_2^2$ as an arithmetic cotangent sum. To analyze $F(x)$ as $x\uparrow 1$, we define the canonical third-order truncation $\mathcal F_{[3]}$ of $F$ by deleting the sole remainder $\rho_3$. We prove exact edge and residue-character cancellations, initial-edge asymptotics, finite-scale formulas, and \[ \mathcal F_{[3]}(x)\ll \frac{\log^2\!\bigl(e/(1-x)\bigr)}{1-x}. \] For the terms containing $\rho_3$, we prove initial-edge asymptotics, and a finite-scale criterion. The unresolved boundedness problem is thereby reduced to explicit global bilinear cancellation.

Review history (2 revisions) →

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Reference graph

Works this paper leans on

19 extracted references · 6 linked inside Pith

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A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis , url =

Báez-Duarte, Luis , journal =. A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis , url =

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1999 , issn =

A Class of Invariant Unitary Operators , journal =. 1999 , issn =. doi:https://doi.org/10.1006/aima.1998.1801 , url =

arXiv 1999

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2002 , url =

Arithmetical versions of Nyman-Beurling Criterion for Riemann Hypothesis , journal =. 2002 , url =

2002

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2002 , author =

New Versions of the Nyman-Beurling Criterion for the Riemann Hypothesis, , journal =. 2002 , author =

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Bettin, Sandro and Conrey, J. Brian , title =. Algebra & Number Theory , volume =. 2013 , doi =. 1111.0931 , archivePrefix =

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Rassias, Michael Th. , title =. Applied Mathematics and Computation , volume =. 2014 , doi =

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Pith/arXiv arXiv 2021

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Vinogradov, I. M. , title =. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya , volume =. 1958 , note =

1958