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REVIEW 3 major objections 2 minor 18 references

On Shusterman's Goldbach-type problem for sign patterns of the Liouville function

T0 review · 3 major / 2 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves, conditional on the Generalised Riemann Hypothesis, that every sufficiently large even integer is a sum of two integers with Liouville value $-1$, via a new lower bound on the frequency of all four sign patterns of the…

desk verdict The paper targets a real open problem with a clever strategy, but the key recursive bound in Proposition 4.1 is arithmetically unsound as written, so the main theorems currently rest on an unjustified estimate. read the letter →

arxiv 2412.17199 v1 pith:3T2GF4IN submitted 2024-12-23 math.NT

Alexander P. Mangerel This is my paper · ORCID
classification math.NT MSC 11N5611N3711M2611P32
keywords LiouvillefunctionbinaryGoldbachproblemsignpatternsGeneralisedRiemannHypothesisDirichletL-functionsPierceexpansionexponentialsumsoversmoothnumberszero-freeregion
topics Goldbach's ConjectureThe Riemann Hypothesis
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

This paper attacks a binary Goldbach analogue: can every large even integer $N$ be written as $a+b=N$ with both $a,b$ having Liouville value $-1$? The author shows that, assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions, the answer is yes for every sufficiently large even $N$. The engine is a quantitative statement for primes: for each large prime $N$, all four sign patterns of $(\lambda(n),\lambda(N-n))$ occur at least $N e^{-C(\log\log N)^6}$ times. This is the first conditional resolution of the sign-pattern version of the problem, and the proof introduces a technique, built on the Pierce expansion of $n/N$, that is designed for pairs whose two entries are at very different scales.

What carries the argument

The argument is carried by the Pierce expansion of the rational number $n/N$, the alternating expansion $n/N=1/r_1-1/(r_1r_2)+1/(r_1r_2r_3)-\cdots$ with strictly increasing integer denominators $r_j$. The paper encodes the failure of an exact dilation symmetry of the Fourier coefficients $S_\lambda(a/N)=\sum_{n<N}\lambda(n)e(na/N)$ through the sets $E_d(N)=\{n<N:\lambda(dn)\lambda(\varphi_d(n))=-1\}$, where $\varphi_d(n)=N\{dn/N\}$. A recursion over Pierce signatures expresses membership in $E_p(N)$ for prime $p$ in terms of smaller $E_r(N)$, at a cost of $O(p|E(N)|)$ exceptions; combined with a reciprocity identity between $E_a(N)$ and $E_b(N)$ under $\varphi$, this controls Fourier coefficients of $E_b(N)$. On the GRH side, averaged prime character sums force some $E_p(N)$ to be large for primes $p$ beyond $(\log N)^{2+\varepsilon}$, and the contradiction between the two estimates yields the sign-pattern lower bound.

What would settle it

Compute, for a large prime $N$, the exact counts of the four sign patterns of $(\lambda(n),\lambda(N-n))$ by factoring all $n<N$; if any pattern is absent, Theorem 1.3's second alternative is false at that $N$. For the proof's engine, locate a modulus $N$ and a non-principal character $\chi$ with an $L$-function zero in the stated rectangle; then the estimate (8) used in Proposition 2.2 would be unavailable.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 1.2: under GRH there is an effectively computable $N_0$ such that every even $N\ge N_0$ admits $1\le a,b<N$ with $a+b=N$ and $\lambda(a)=\lambda(b)=-1$. The result is derived from the stronger Proposition 1.4, which asserts that for every sufficiently large prime $N$ and every pattern $(\eta_1,\eta_2)\in\{-1,+1\}^2$ there are $\gg N e^{-C(\log\log N)^6}$ integers $n<N$ with $\lambda(n)=\eta_1$ and $\lambda(N-n)=\eta_2$. The author is careful to record that full GRH is not needed: a zero-free rectangle of the form $\operatorname{Re}(s)>1-(\log N)^{-c}$ with $c<3/50$ and $|\operatorname{Im}(s)|\le(\log N)^3$ would suffice, though with a weaker exponent.

Load-bearing premise

The load-bearing premise is that no non-principal Dirichlet character modulo $N$ has a zero in the rectangle $\operatorname{Re}(s)>1-(\log N)^{-c}$, $|\operatorname{Im}(s)|\le(\log N)^3$ with $c<3/50$; if such a zero exists, the averaged prime character sum estimate (8) fails and the contradiction cannot be run.

Editorial extensions

If this is right

  • Under GRH, every sufficiently large even integer is a sum of two positive integers each having an odd number of prime factors counted with multiplicity.
  • For every sufficiently large prime $N$, all four sign patterns of the pair $(\lambda(n),\lambda(N-n))$ appear, and each appears many times: $\gg N e^{-C(\log\log N)^6}$.
  • The evenness constraint in the binary-Goldbach analogue is only needed for the small friable residual cases; for any integer $N$ that has a prime factor larger than a computable $p_0$, all four sign patterns occur.
  • Because $N_0$ is effectively computable in terms of the parameter $p_0$, the remaining finitely many even $N$ below the threshold are in principle amenable to numerical verification.
  • The proof's zero-free-region requirement is much weaker than GRH, so the same theorem would hold under any future proof of the stated rectangle.

Reading between the lines

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

  • If the appendix's average estimate for the preimage count $\nu_r(m)$ could be upgraded to a typical or second-moment bound, the recursion would allow a much larger smoothness parameter $q$ and the contradiction would only need a Littlewood-type zero-free region; an unconditional theorem might then be within reach of zero-density methods.
  • The approximate dilation symmetry and Pierce-expansion encoding are not tied to $\lambda$: the same architecture could plausibly establish sign-pattern frequencies for other real-valued multiplicative functions whose non-pretentiousness is available, especially in two-variable sums where the summands have very different sizes.
  • A direct computational check on primes $N$ up to any feasible bound, using exact values of $\lambda$, would either confirm Proposition 1.4 in that range or, if a sign pattern is missing, give a concrete counterexample to the unconditional content of Theorem 1.3 for that $N$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 2 minor

Summary. The manuscript assumes GRH and claims that every sufficiently large even integer N has a representation N = a + b with λ(a) = λ(b) = -1, answering a problem of Shusterman. The main route is a quantitative sign-pattern count for prime N: Proposition 1.4 asserts that each of the four sign patterns occurs at least ≫ N exp(-C(log log N)^6) times. The proof proceeds by studying the minimal exceptional set E(N), proving an approximate dilation symmetry for Fourier coefficients of λ (Proposition 2.1), and contradicting it on average over primes via character sums under GRH (Proposition 2.2). The key intermediate is a recursive bound on the exceptional sets E_d(N) via Pierce expansions (Proposition 4.1), which is then combined with reciprocity (Lemma 4.3) and exponential sums over smooth numbers (Proposition 5.1).

Significance. If the technical content were correct, this would be a significant contribution: it would give a conditional answer to a natural analogue of the binary Goldbach problem under GRH, with an explicit quantitative lower bound for all sign patterns when N is prime. The proposed use of Pierce expansions and approximate dilation symmetry is novel and potentially useful for other binary problems. The paper is also careful about effectivity and about the possibility of weakening GRH to a zero-free region. However, the central recursive estimate, Proposition 4.1, is not established as written, and because the later Fourier-analytic argument depends on it, the main theorems are not proved by this manuscript.

major comments (3)
  1. [Section 4, proof of Proposition 4.1, Eq. (14)] The claimed induction is arithmetically false. The displayed recurrence g(p) ≤ 2p + Σ_{1≤r<p} 2^{r-1} g(r), combined with the induction hypothesis g(r) ≤ 2r^2, yields a right-hand side of size about 2^p p^2, not O(p^2). For example, with g(1)=0, g(2)≤2, g(3)≤6 and g(4)≤8 (the last from Lemma 4.5), the recurrence gives g(5)≤102, which contradicts the claimed g(5)≤50. Thus the universal bound g(d)≤2d^2 is unsupported.
  2. [Section 4, Eq. (12)] The composite step of the induction is also invalid as displayed. Substituting Ω(R)≤2 log R and P^+(R)≤R/2 gives Ω(R)^2 2P^+(R)^2 ≤ 2R^2(log R)^2, which is not ≤ 2R^2 for R≥4. Even if the prime case were repaired, the composite case would require a separate argument or a weaker allowed bound.
  3. [Sections 5 and 6.1 (Lemma 5.2, Proposition 5.1, proof of Proposition 1.4)] Because Lemma 5.2 and Proposition 5.1 rely directly on Proposition 4.1 for the size of |E_a(N)| with q-friable a, the failure of Proposition 4.1 invalidates the exponential-sum estimate (15) and hence Proposition 2.1. The contradiction in the proof of Proposition 1.4 depends on Proposition 2.1, so the main theorems do not follow. The appendix's average bound on ν_r(m) cannot repair Eq. (14), since the induction needs pointwise control of ν_r(m) on the potentially sparse subset φ_p^{-1}(E_r(N))∩(N/(r+1),N/r), not merely on the full interval.
minor comments (2)
  1. [Proposition 1.4] The last line of the proposition reads 'there are 3 ≫ N e^{-C(log log N)^6} integers'; the symbol '3' appears to be a typo and should be '≥' or '≫'.
  2. [Throughout] The typeset text frequently collapses superscripts in the key estimates (for example '2(1+o(1))q2' and '22q2'), making it difficult to verify exact exponents; the manuscript should be typeset with unambiguous superscripts.

Circularity Check

0 steps flagged · score 2.0 of 10

Score 2: no circularity. Theorem 1.2/Prop 1.4 rest on a contradiction between a recursion upper bound (Prop 2.1) and a GRH-conditional lower bound (Prop 2.2) on |Ep(N)|; neither presumes the target. Self-citations to [7] are published, parameter-free, strictly weaker inputs — legitimate support.

full rationale

Derivation chain: Theorem 1.2 follows from Theorem 1.3, which follows from Proposition 1.4; its second claim is an algebraic identity in lambda-values (Section 6.1), and the first claim is proved by contradiction. Assuming |E(N)| < Ne^{-C(log log N)^6}, Proposition 2.1 (Pierce-expansion recursion, Sections 4-5) makes the second moment (1/N)Sum|Slambda(pa/N)-lambda(p)Slambda(a/N)|^2 = 4|Ep(N)|/N (Lemma 3.1) small for p in (P/2,P], while Proposition 2.2 (GRH/zero-free-region lower bound via character orthogonality, Lemmas 3.2-3.3) forces some |Ep(N)| >> N. These bounds concern the same quantity but derive from independent inputs; no parameter is fitted, no quantity is defined in terms of the (-,-)-pattern claim, and the zero-free region (Remarks 1 and 9) is an external hypothesis. Structural circularity is absent. Self-citations to [7]: (i) E(N) nonempty from [7] makes g(d) := |Ed(N)|/|E(N)| well-defined (Section 4); (ii) the identity Lp(n) = Lr(phip(n))Lp(theta(n)) is cited from [7, Lem. 3.1(d)] (Section 4.3); (iii) [7, Thm. 1] supplies the auxiliary same-sign pair for odd M >= 11 in case (iv) of Theorem 1.2. Each is published, parameter-free, and strictly weaker than the target, hence genuine evidence that does not raise the circularity score. No uniqueness theorem is imported; the dilation symmetry (3) is re-derived, not assumed. Flagged per the review rule: the paper records its own fragility (Remark 7: 'we are forced to take q = O(sqrt(log N))'; footnote 12: 'an error in this estimate in a previous version'), and the printed inequalities are violated. From (14), g(p) <= 2p + Sum_{1<=r<p} 2^{r-1}g(r); even granting g(r) <= 2r^2, the sum is of order 2^p p^2, so the displayed chain '2p + 2(p-1)^{2+p-2} + (p-2)2^{(p-2)^2+p-3} < 2p^2' is numerically false (the recurrence gives g(5) <= 102 > 50), and (12) drops a factor (log R)^2: '(2 log R)^2 2(R/2)^2 <= 2R^2' would require log R <= 1. In the proof of Proposition 1.4, '2^{(1+o(1))q^2}|E(N)| = o(N)' with q = (C/10)(log log N)^6 and |E(N)| < Ne^{-C(log log N)^6} is false: 2^{q^2}|E(N)| >> N. These failures leave the quantitative engine unproved as written; the Appendix's average bound on nu_r(m) does not repair the pointwise 2^{r-1} loss. This is a correctness risk, not circularity: the broken bounds are asserted, not assumed. Score 2 reflects multiple but legitimate self-citations; the central claim retains independent content.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The load is carried by GRH, standard analytic number theory tools, and the author's own earlier paper [7].

assumptions (6)
  • domain assumption GRH for Dirichlet L-functions, or the weaker zero-free region of Remark 1
    Assumed in Theorems 1.2, 1.3 and Proposition 1.4; enters through Lemma 3.3 and Proposition 2.2 to control averaged prime character sums.
  • domain assumption Main theorem of Mangerel [7] giving E(N) nonempty and the existence of n < M/2 with lambda(n) = lambda(M-n) for odd M
    Used in Section 4 to define g(d) = |E_d(N)|/|E(N)| and in Theorem 1.2 case (iv) for the friable case.
  • standard math Harper's exponential sum bound for smooth numbers
    Quoted in Section 5 and used in Proposition 5.1 to control Fourier coefficients supported on E_b(N).
  • standard math Fouvry-Tenenbaum Theorem 13 exponential sum bound
    Used in Proposition 5.1 for the range not covered by Harper's theorem.
  • standard math Canfield-Erdos-Pomerance estimate for Psi(T,q)
    Used in Proposition 5.1 to lower-bound the count of q-smooth numbers up to T.
  • standard math Prime number theorem with Vinogradov-Korobov error
    Used in Proposition 1.4 to bound the sum of lambda(n) and in Lemma 3.3.

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Pith. "Pith review of On Shusterman's Goldbach-type problem for sign patterns of the Liouville function." pith.science (2026). https://pith.science/paper/3T2GF4IN

@misc{pith2026241217199,
  author       = {Pith},
  title        = {Pith review of: On Shusterman's Goldbach-type problem for sign patterns of the Liouville function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3T2GF4IN}},
  note         = {Machine review of arXiv:2412.17199}
}
abstract

Let $\lambda$ be the Liouville function. Assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions (GRH), we show that for every sufficiently large even integer $N$ there are $a,b \geq 1$ such that $$ a+b = N \text{ and } \lambda(a) = \lambda(b) = -1. $$ This conditionally answers an analogue of the binary Goldbach problem for the Liouville function, posed by Shusterman. The latter is a consequence of a quantitative lower bound on the frequency of sign patterns attained by $(\lambda(n),\lambda(N-n))$, for sufficiently large primes $N$. We show, assuming GRH, that there is a constant $C > 0$ such that for each pattern $(\eta_1,\eta_2) \in \{-1,+1\}^2$ and each prime $N \geq N_0$, $$ |\{n < N : (\lambda(n),\lambda(N-n)) = (\eta_1,\eta_2)\}| \gg N e^{-C(\log \log N)^{6}}. $$ The proof makes essential use of the Pierce expansion of rational numbers $n/N$, which may be of interest in other binary problems.

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