Multiple sums with the M\"obius function

Igor E. Shparlinski; William D. Banks

REVIEW 1 major objections 3 minor 18 references

Multiple sums with the M\"obius function

T0 review · 1 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that sums of the Möbius function over solutions to a broad class of ternary additive equations are bounded by (A+B)N divided by any fixed power of log N, matching the strength of the error-term bounds known for the…

desk verdict Clean new bounds for ternary Möbius sums, resting on Green-Tao; worth a serious referee. read the letter →

arxiv 2506.08787 v1 pith:6DFFHCAT submitted 2025-06-10 math.NT

William D. Banks , Igor E. Shparlinski This is my paper

classification math.NT MSC 11P3211N5611L0711B30

keywords Goldbach-typetheoremsMöbiusfunctionDavenporttheorembracketpolynomialsexponentialsumsternaryadditiveequationsshortintervalsnilsequences

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes uniform cancellation bounds for the Möbius function over solutions to a broad family of ternary additive equations. For intervals $A=[1,A]$, $B=[1,B]$, $N=[1,N]$ with $\log N \ge (\log \max\{A,B\})^\varepsilon$, and for injective functions $f,g$ and any bracket polynomial phase $\wp$ of complexity $\delta$, the sum of $\mu(n_1n_2n_3)$ over triples with $f(n_1)+g(n_2)+\wp(n_3)=M$ is bounded by $(A+B)N(\log N)^{-C}$ for every fixed $C>0$. This matches the strength of the best error-term bounds for the ternary Goldbach problem with primes, and it is new even for the plain ternary sum $\sum_{n_1+n_2+n_3=N}\mu(n_1n_2n_3)$, which is shown to be $\ll N^2(\log N)^{-C}$. The paper also proves a short-interval analogue with $H\ge N^{5/8+\varepsilon}$ and conditional improvements assuming zero-free half-planes or the absence of Siegel zeros for Dirichlet $L$-functions, with applications to primes, squarefree products, Beatty sequences, and quadratic residues.

What carries the argument

The engine of the proof is the strong discorrelation theorem for the Möbius function against bracket polynomial phases, stated in the paper as (3.2): for any bracket polynomial $p$ of complexity $\delta$ and any $1$-Lipschitz $\Psi\colon[0,1]\to[-1,1]$, one has $\sum_{n\le N}\mu(n)\Psi(\{p(n)\})\ll_{\delta,\Psi,C} N(\log N)^{-C}$, with an ineffective constant. This is a theorem proved elsewhere, and the paper extends it to arithmetic progressions in Lemma 3.1 by orthogonality. The proof of Theorem 2.1 expands the equation $f(n_1)+g(n_2)+\wp(n_3)=M$ via exponentials, uses the discorrelation bound to estimate the sum over $n_3$, and uses the injectivity of $f$ and $g$ together with Cauchy-Schwarz to control the sums over $n_1,n_2$; a divisor split into small and large $d$ balances the error terms. A bracket polynomial is an expression built from scalars, $+$, $\times$, integer part $\lfloor\cdot\rfloor$, and fractional part $\{\cdot\}$, with complexity the minimal number of such operations needed to write it.

What would settle it

A single counterexample would falsify the main theorem: find a bracket polynomial $p$ and a sequence of $N$ for which $\sum_{n\le N}\mu(n)e(p(n))$ is not $O(N(\log N)^{-C})$ for some fixed $C$; the proof of Theorem 2.1 uses exactly this bound (via Lemma 3.1) to estimate the $n_3$ sum. Short of that, a direct numerical computation of $S_\mu(N)=\sum_{n_1+n_2+n_3=N}\mu(n_1n_2n_3)$ that shows $\lvert S_\mu(N)\rvert N^{-2}(\log N)^C$ unbounded for some $C$ would contradict (2.1).

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a uniform bound for the sums $S_\mu(\mathcal{D};M)$ defined in (1.5): under the hypotheses of Theorem 2.1, $S_\mu(\mathcal{D};M) \ll_{C,\varepsilon,\delta} (A+B)N(\log N)^{-C}$ for every $C>0$, uniformly in $M \in \mathbb{Z}$. The bound holds for arbitrary complex weights $u_n,v_n$ with sup norm at most $1$, for injective functions $f$ and $g$ on the supports of the weights, and for any bracket polynomial $\wp$ of complexity $\delta$. A direct consequence, displayed in (2.1), is the new estimate $\sum_{n_1+n_2+n_3=N}\mu(n_1n_2n_3)\ll_C N^2(\log N)^{-C}$ for the ordinary ternary Möbius sum. The paper further shows that, when $\wp$ is linear, stronger conditional bounds follow from hypotheses on the zeros of Dirichlet $L$-functions (Theorem 2.2), and that a short-interval version holds for ordinary polynomial phases with $H\ge N^{5/8+\varepsilon}$ (Theorem 2.4).

Load-bearing premise

The proof rests on the discorrelation theorem for the Möbius function over bracket polynomial phases, stated in the paper as (3.2) and proved elsewhere; if that theorem failed for some bracket polynomial phase, the main cancellation bound would fail for that phase.

Editorial extensions

If this is right

The plain ternary sum satisfies $\sum_{n_1+n_2+n_3=N}\mu(n_1n_2n_3)\ll_C N^2(\log N)^{-C}$, a new estimate with the same logarithmic strength as the known error term for the ternary Goldbach problem.
Prime applications include $\sum_{p+q\le x}\mu(p+q)\ll_C x^2(\log x)^{-C}$ and a signed sum of $\mu(p_k+p_\ell)$ over prime indices, both new.
When the third variable is confined to a short interval of length $H\ge N^{5/8+\varepsilon}$, the same cancellation holds for ordinary polynomial phases, giving $\ll_{C,\varepsilon,\delta}(A+B)H(\log N)^{-C}$.
Restricting the third variable to $H\ge \exp((\log N)^\varepsilon)$ gives $\sum_{n_1+n_2+n_3=N,\,n_3\le H}\mu(n_1n_2n_3)\ll_{\varepsilon,C} NH(\log H)^{-C\varepsilon^{-1}}$, a partial step toward the Chowla-type one-variable problem.
Under the hypothesis that no Siegel zeros exist for Dirichlet $L$-functions, the bound improves to $(A+B)N e^{-c\sqrt{\log N}}$ for linear $\wp$; under a zero-free half-plane, to $(A+B)N^{c(\sigma^*)+o(1)}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extending beyond the paper's own claims: the method appears to transfer to any multiplicative function whose discorrelation against nilsequences is known (such as the Liouville function), provided the same coprimality handling applies, which would yield analogous cancellation with $\lambda$ in place of $\mu$.
The injectivity of $f$ and $g$ is used so that the $n_1,n_2$ sums become $L^2$ norms; numerically testing non-injective perturbations such as $f(n)=\lfloor n/2\rfloor$ would probe whether this condition is truly necessary.
The one-small-variable bound, requiring $H\ge \exp((\log N)^\varepsilon)$, leaves an exponential gap to the $H=1$ Chowla-type case; a reasonable testable conjecture is that the threshold on $H$ can be pushed to a power of $\log N$, which numerical experiments at moderate $N$ could inform.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The paper is a good, honest piece of analytic number theory. The main result (Theorem 2.1) is a uniform, logarithmically strong bound for sums of Möbius over solutions to f(n1)+g(n2)+℘(n3)=M, with f and g injective and ℘ a bracket polynomial. The proof is short and clean: inclusion-exclusion, orthogonality, Cauchy-Schwarz, and the Green-Tao discorrelation bound (3.2). The new instances are real: (2.1) for the plain ternary Möbius sum, the quadratic residue application in §7.5, and the small-variable range H ≥ exp((log N)^ε) in §7.6. That last one improves on what is known for the analogous von Mangoldt sums, where only much larger H are handled.

The soft spots are the external dependencies. The entire engine is the Green-Tao theorem, a deep input with an ineffective constant. The paper does not reprove it, and the uniformity in Lemma 3.1 depends on the (correct) uniformity of the constant in (3.2) in the phase coefficients. That is standard and the cited theorem does supply it, so I do not see a flaw. Some applications, like (7.1), overlap with what a classical circle method plus Perron can do; the authors say so themselves. The divisor estimates are elementary and correct. The large-d counting in §4.3 is sound: from d | n1 n2 one gets a divisor ≥ sqrt(d) dividing one of the two variables, and injectivity closes the count.

The paper is honest about limitations: the binary case and H=1 are explicitly out of reach, and the conditional results (Theorem 2.2) and conjectures are clearly separated. Citation pattern is appropriate; no self-citation issues.

Who is this for? Analytic number theorists working on Möbius orthogonality, Goldbach-type problems, or quantitative Sarnak. It deserves a serious referee. The proof can be checked in an afternoon; the main risk is the external theorem, not the internal argument. Send it to review.

Referee Report

1 major / 3 minor

Summary. The paper proves uniform cancellation bounds for sums of the Möbius function over solutions to three-variable additive equations, where two variables are weighted by bounded sequences attached to injective label functions and the third runs over an interval and is inserted through a bracket polynomial phase. The main result (Theorem 2.1) gives a bound of size (A+B)N (log N)^{-C} under the mild separation condition log N >= (log max{A,B})^epsilon. The proof splits by a parameter D, using Green-Tao discorrelation for small d and a divisor-counting/injectivity argument for large d. The paper then derives stronger conditional bounds under zero-free half-plane hypotheses (Theorem 2.2), a short-interval variant (Theorem 2.4), and a series of applications to prime, squarefree, Beatty, and quadratic-residue sums, including the new bound S_mu(N) << N^2 (log N)^{-C} for the ternary Möbius sum.

Significance. If correct, the paper makes a solid contribution to the analytic theory of the Möbius function. It shows that the deep Green-Tao discorrelation theorem can be converted, by a clean and relatively short argument, into cancellation for a broad family of additive equations, matching the strength of the best known error terms in the ternary Goldbach problem. The paper is explicit about its external inputs: the main engine is the Green-Tao bound (3.2) and its arithmetic-progression variant (Lemma 3.1), both cited rather than reproved, and the implied constants are honestly ineffective. The applications are numerous and several are new. The proof is transparent; the split at a parameter D and the use of divisor-function estimates (3.10)-(3.13) are standard and correctly executed. The only notable weakness is a misstated application in §7.3, discussed below, which is local and does not affect the validity of Theorem 2.1.

major comments (1)

[§7.3] In the application labelled 'Squarefrees', the data specify N = [1,x], but the displayed sum restricts k+ℓ to the interval (x,x+H]. Since the equation f(k)+g(ℓ)+℘(n3)=0 with ℘(X)=-X forces n3 = k+ℓ, and n3 ∈ N = [1,x], no tuples satisfy x<k+ℓ≤x+H. The displayed estimate is therefore vacuous as stated. To obtain a nontrivial short-interval result one would need to take N = (x,x+H] and invoke Theorem 2.4 (with the accompanying lower bound on H), rather than Theorem 2.1; alternatively, the range should be changed to k+ℓ≤x. Please correct this application.

minor comments (3)

[Lemma 3.6] In the statement of Lemma 3.6, 'for all A>0' should read 'for all C>0'; the bound contains C, not A. Also, 'be an s an ordinary polynomial' contains a typo.
[§7.1] The condition on A is stated as 'A ≤ e^{(log x)^C}' and later A is taken of size x. For arbitrary C>0 this requires choosing the exponent in the theorem's application to be at least 1; the authors may wish to clarify that one first applies Theorem 2.1 with a larger exponent and then passes to the desired C.
[§4.2] In the proof of Theorem 2.1, the use of Lemma 3.1 for the inner sum over n3 is uniform in the integration variable α because multiplication by a scalar is allowed in the bracket-polynomial calculus; it would be helpful to state this explicitly for the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is derived from the external Green-Tao discorrelation theorem and elementary divisor estimates, with no self-referential reduction.

full rationale

The derivation chain of Theorem 2.1 is: express S_mu(D;M) via orthogonality, decompose by divisibility d, bound the small-d part using the arithmetic-progression discorrelation bound of Lemma 3.1, and bound the large-d part by counting with the injectivity of f and g. Lemma 3.1 is proved in the paper from (3.3), a direct consequence of the cited Green-Tao bound (3.2) [7, Thm. 5.2], by expanding the congruence condition; the uniformity in d and a follows because each phase p(n)+jn/d has complexity O(delta) and the implied constant in (3.3) depends only on delta. The small-d estimate then uses standard divisor-function moment bounds (3.10)-(3.13). None of these inputs asserts the conclusion S_mu(D;M) << (A+B)N(log N)^-C; they are independent results by other authors or elementary estimates. The applications in Section 7 instantiate Theorem 2.1 with particular choices of data; they do not re-fit any parameter. There are no self-citations of the authors' own prior work that carry a load-bearing premise, and no fitted quantity is relabelled as a prediction. The only notable feature is the ineffective constant inherited from the Green-Tao theorem, which is an external limitation, not circularity. Theorem 2.2 similarly relies on external discorrelation results of Hajela-Smith, Baker-Harman, and Zhang, and Theorem 2.4 relies on the external short-interval result [12, Cor. 1.3(i)]. The paper is therefore self-contained in the relevant sense: its central claims are derived from external benchmarks, not from the target conclusions.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities; it uses known objects (Möbius function, bracket polynomials, Dirichlet L-functions). The free parameters are arbitrary constants in the theorems, and the axioms are the deep inputs the proof relies on, which are standard in the field. The central claim does not pull a new particle, force, or dimension from a hat.

free parameters (2)

C
C is an arbitrary fixed positive real in Theorems 2.1, 2.2, 2.4; it is not fitted but appears in the exponent of the logarithm, and the bound holds for every C.
epsilon
epsilon is an arbitrary fixed positive real in Theorem 2.1, appearing in the condition log N >= (log max{A,B})^epsilon; not fitted.

assumptions (4)

domain assumption Green-Tao theorem on strong orthogonality of the Möbius function to bounded bracket polynomial phases (Eq. 3.2).
The paper does not prove this; it is a deep 2012 theorem by Green and Tao, referenced in the paper, and is the core input for Lemma 3.1 and hence for the whole paper.
domain assumption Matomaki-Shao-Tao-Teravainen short-interval polynomial discorrelation (Eq. 3.9).
Used in the short-interval Theorem 2.4 via Lemma 3.6, and cited as Cor. 1.3 of [12].
standard math Standard analytic number theory estimates: divisor function mean values (3.10), partial summation consequences (3.11), (3.12), and elementary bound (3.13).
These are classical and elementary, and the paper sketches proofs for some.
domain assumption Uniformity of the implied constant in Lemma 3.1 independent of the progression a mod d.
This is a consequence of the Green-Tao bound and orthogonality, but the paper states it in Remark 3.2 without a full proof; it is a load-bearing assumption for summing over d.

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Cite this review

Pith. "Pith review of Multiple sums with the M\"obius function." pith.science (2026). https://pith.science/paper/6DFFHCAT

@misc{pith2026250608787,
  author       = {Pith},
  title        = {Pith review of: Multiple sums with the M\"obius function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6DFFHCAT}},
  note         = {Machine review of arXiv:2506.08787}
}

read the original abstract

We establish nontrivial bounds for bilinear sums involving the M\"obius function evaluated over solutions to a broad class of equations. Several of our results may be regarded as M\"obius-function analogues of the ternary Goldbach problem. By contrast, the binary versions of our results remain out of reach, much like the binary Goldbach problem. Nevertheless, we make partial progress in this direction by restricting the range of the third variable as far as possible.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

R. C. Baker and G. Harman, Exponential sums formed with the M¨ obius function, J. London Math. Soc. (2)43(1991), no. 2, 193–198. MR1111578 (p. 7)

work page 1991

[2] [2]

Bergelson and A

V. Bergelson and A. Leibman, Distribution of values of bounded generalized polynomials, Acta Math.198(2007), no. 2, 155–230. MR2318563 (p. 3)

work page 2007

[3] [3]

Cai, A remark on the Goldbach-Vinogradov theorem, Funct

Y. Cai, A remark on the Goldbach-Vinogradov theorem, Funct. Approx. Comment. Math. 48(2013), no. 1, 123–131. MR3086965 (pp. 2 and 18)

work page 2013

[4] [4]

Davenport, On some infinite series involving arithmetical functions II, Quart

H. Davenport, On some infinite series involving arithmetical functions II, Quart. J. Math. 8(1937), 313–320, DOI 10.1093/qmath/os-8.1.8. (p. 6) MULTIPLE SUMS WITH THE M¨OBIUS FUNCTION 19

work page doi:10.1093/qmath/os-8.1.8 1937

[5] [5]

Fouvry and S

´E. Fouvry and S. Ganguly, Strong orthogonality between the M¨ obius function, additive characters and Fourier coefficients of cusp forms, Compos. Math.150(2014), no. 5, 763–

work page 2014

[6] [6]

Gafni and N

A. Gafni and N. Robles, Exponential sums weighted by additive functions, Preprint, 2025, (available fromhttps://www.arxiv.org/abs/2502.05298). (p. 2)

work page arXiv 2025

[7] [7]

B. J. Green and T. C. Tao, The M¨ obius function is strongly orthogonal to nilsequences, Ann. of Math. (2)175(2012), no. 2, 541–566. MR2877066 (pp. 3, 6, and 8)

work page 2012

[8] [8]

Hajela and B

D. Hajela and B. P. Smith, On the maximum of an exponential sum of the M¨ obius function, Number theory (New York, 1984–1985), 145–164, Lecture Notes in Math., 1240, Springer, Berlin. MR0894510 (p. 7)

work page 1984