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REVIEW 4 major objections 6 minor 17 references

Goldbach's Problem in short intervals for numbers with a missing digit

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Pith's one-line read Assuming a zero-free region for Dirichlet L-functions, almost all even numbers with a missing digit in short intervals are sums of two primes.

desk verdict New short-interval Goldbach result for missing-digit numbers, but the core minor-arc bound is a one-line citation transfer that needs verification. read the letter →

arxiv 2412.19975 v1 pith:5ZQSES4E submitted 2024-12-28 math.NT

Jiseong Kim This is my paper · ORCID
classification math.NT MSC 11P3211N0511N3711A63
keywords GoldbachnumbersmissingdigitsshortintervalsDirichletL-functionszero-freeregiondivisorfunctioncirclemethodexponentialsums
topics Goldbach's ConjectureThe Twin Prime Conjecture
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 studies even numbers in a short interval whose base-$g$ expansion omits one fixed digit, and asks whether they are Goldbach numbers, i.e., sums of two primes. The main theorem says that, assuming a zero-free region for Dirichlet $L$-functions, almost all even $m \in [X,X+H]^*$ are Goldbach numbers whenever $H = X^{2/3+\varepsilon}$ (with $H$ also at most $X^{1-\varepsilon}$); the exceptions have size $o(|[X,X+H]^*|)$. A supporting, unconditional result bounds the average of the fourth divisor function $d_4(n)$ over the missing-digit interval by $O((\log X)^7 |[X,X+H]^*|)$. The result matters because missing-digit sets are sparse and have almost no multiplicative structure, so the usual sieve and circle-method tools do not apply directly; the Fourier transform of the digit constraint is what carries the argument. The paper therefore extends the short-interval Goldbach phenomenon to these digit-restricted sets, conditional on the zero-free region and on the imported estimates used to control the minor arcs.

What carries the argument

The machinery is the circle method run on the missing-digit interval. Two families of exponential sums do the work: the weighted prime sums $S_1(\alpha), S_2(\alpha)$ over the intervals $I_1=(X-H,X]$, $I_2=(0,H]$, and the missing-digit exponential sum $F(\alpha)=|[X,X+H]^*|^{-1}\sum_{n\in[X,X+H]^*} e(n\alpha)$. The circle is split into Farey arcs (intervals around rationals $a/q$ with $q \le Q=X^{\varepsilon}$); on the major arcs the main term is the singular series $\sum_q \mu(q)^2\varphi(q)^{-2} c_q(-2n)$ times the trivial convolution count, and the error terms are bounded using the zero-density estimate for Dirichlet $L$-functions and the assumed zero-free region. On the minor arcs, the bound for $S_i(\alpha)$ is imported from earlier work on primes in short intervals and polynomial phases, and the average size of $F(\alpha)$ is controlled by an $L^1$-average estimate for the missing-digit exponential sum. That $L^1$ estimate, together with a type-I decomposition of a smoothed $d_4$ approximant, is the mechanism by which the sparse digit-restricted set behaves like a short interval.

What would settle it

If a direct check of Lemma 1.6 with $H=X^{2/3+\varepsilon}$ and $\delta=X^{-\varepsilon}$ finds even one $\alpha$ on a minor arc (far from every rational with denominator at most $X^\varepsilon$) for which $\max(S_1(\alpha),S_2(\alpha))$ exceeds $H X^{-c_3\varepsilon}$, then the minor-arc integral in the proof is not controlled and the paper's proof of Theorem 1.2 collapses.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.2: if some fixed $c_2>0$ has the property that $L(s,\chi)$ never vanishes in $\Re(s)>1-c_2$ for any Dirichlet character $\chi$, then for $X^{2/3+\varepsilon} \ll H \ll X^{1-\varepsilon}$ and base $g$ sufficiently large, almost all even numbers $m \in [X,X+H]^*$ are Goldbach numbers; equivalently, the set of even exceptions has size $o(|[X,X+H]^*|)$. The proof follows the classical circle-method scheme for Goldbach in short intervals: the unit circle is split into Farey arcs, the major arcs give the main term $\mathfrak{S}(-2n) M^*(2n)$ built from Ramanujan sums, and the minor arcs are controlled by combining an upper bound for the weighted prime exponential sums with an $L^1$-average bound for the missing-digit exponential sum. A supporting result, Theorem 1.1, is unconditional: for $X^{3/5+\varepsilon} \ll H \ll X^{1-\varepsilon}$, the average of $d_4(n)$ over $[X,X+H]^*$ is $O((\log X)^7 |[X,X+H]^*|)$. This divisor bound is used to show that the average of a Ramanujan-sum tail in the major-arc analysis is $o(|[X,X+H]^*|)$, which is what makes the 'almost all' conclusion quantitative.

Load-bearing premise

The load-bearing assumption is that a minor-arc bound on sums over primes proved in earlier work, for slightly different parameters, still holds after the paper replaces the smoothing parameter by $\delta=X^{-\varepsilon}$ and imposes $H\ge X^{2/3+\varepsilon}$; the paper does not verify that the constants survive this substitution, and if they do not, the minor-arc contribution is not controlled and the theorem is not established.

Editorial extensions

If this is right

  • Under the zero-free-region assumption, the set of even $m \in [X,X+H]^*$ that are not Goldbach numbers has size $o(|[X,X+H]^*|)$
  • The unconditional divisor estimate gives an average order for $d_4(n)$ on missing-digit intervals with only a logarithmic loss, matching what is known for ordinary short intervals up to the exponent of $\log X$
  • The main-term shape $\mathfrak{S}(-2n) M^*(2n)$ shows the Goldbach asymptotic inside the digit-restricted interval has the same singular-series structure as the classical Goldbach problem
  • The combination of the type-I decomposition and the missing-digit Fourier bound is what makes the minor-arc integral $o(H |[X,X+H]^*|)$, so the same two inputs should control other binary additive problems over the same sets

Reading between the lines

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

  • Beyond the paper's claims, the same circle-method template should work for any sparse digit-constrained set whose exponential sum has small $L^1$ average (on the order of $N^{-1+o(1)}$), suggesting a general principle linking small Fourier dimension to short-interval Goldbach phenomena
  • The unconditional $d_4$ bound is likely to be reused independently, for example in shifted divisor sums or correlations over missing-digit sets that currently rely on weaker pointwise estimates
  • A computational check of Lemma 1.6 for moderate $X=g^k$, $H=X^{2/3+\varepsilon}$, and a few exclusions $b$ would reveal whether the imported minor-arc constants are plausible; a violation on the complement of the Farey arcs would point to the exact repair needed
  • The paper does not address what happens below $H=X^{2/3+\varepsilon}$; determining whether that threshold is an artifact of the imported estimates or an intrinsic feature of the digit set would clarify the limits of the method
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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

4 major / 6 minor

Summary. The manuscript studies Goldbach's problem over integers in a short interval [X,X+H] that have a missing base-g digit. Assuming a fixed zero-free region for Dirichlet L-functions, it claims (Theorem 1.2) that for H ≥ X^{2/3+ε} and g sufficiently large, almost all even elements of [X,X+H]^* are Goldbach numbers. The proof uses the Hardy–Littlewood circle method: major arcs are treated via zero-density estimates and the zero-free-region assumption (Lemma 1.5), the prime exponential sums on minor arcs are bounded by a transfer from Matomäki–Shao (Lemma 1.6), and the missing-digit set is handled through an L1-average estimate for its Fourier sum (Lemma 1.7). A supporting result, Theorem 1.1, bounds the fourth divisor function over [X,X+H]^* by O((log X)^7 |[X,X+H]^*|) for H ≥ X^{3/5+ε}, and is proved by approximating d4 via type-I sums and using a similar major/minor-arc decomposition. The paper is conditional on an external unproved hypothesis (a zero-free region for Dirichlet L-functions), but it also contains several internal inputs that are asserted without proof.

Significance. If the gaps were filled, the result would be a natural and valuable extension of the Perelli–Pintz short-interval Goldbach theorem to digit-restricted numbers, combining recent developments in short-interval prime correlations with Fourier analysis of missing-digit sets. The overall strategy is coherent, and the major-arc computation in Theorem 1.2 is standard. The paper also gives credit to the recent machinery of Matomäki–Shao and Matomäki–Radziwiłł–Shao–Tao–Teräväinen; a complete proof would be a useful contribution. However, as submitted the central claims rest on several unproved or merely cited inputs—notably Lemma 1.6, Lemma 1.7, and Proposition 3.5—so the significance can only be assessed after those gaps are closed. The manuscript is not currently self-contained enough to verify the headline theorem.

major comments (4)
  1. [§1.2, Lemma 1.6] This lemma is the decisive minor-arc estimate in Theorem 1.2, and its proof is a one-sentence delegation to [12] ('The proof comes from the argument in [12]. The only difference is, we use δ as a power of X^{-ε}'). No precise statement from [12] is quoted, and the manuscript does not verify that the hypotheses of that theorem hold for H = X^{2/3+ε}, Q = X^ε, and β(δ) = (log X)^{37}/(Hδ). In particular, it is not shown that H lies in the range required by the cited result, that the Farey-arc width β(δ) is compatible with the cited result's constants, or that the saving c3 can be chosen uniformly in g. Since the minor-arc contribution in Section 1.3 is controlled exactly by this bound, the claimed conclusion o(H|[X,X+H]^*|) is not established unless Lemma 1.6 is proved in detail.
  2. [§1.2, Lemma 1.7] Lemma 1.7 is an unproved assertion about the L1-average of the normalized missing-digit Fourier sum F_{[X,X+H]}. It is used in Proposition 2.2 (to replace S4 by S4^♯), in Proposition 2.5 (to control the minor arcs in the d4 argument), and in the final minor-arc estimate of Section 1.3. The proof is absent, and no reference is given. The statement is nontrivial: the exponent log(log g+1)/log(g-1) is not derived. Furthermore, the final o(H|[X,X+H]^*|) conclusion requires this exponent to be smaller than 2c3ε, where c3 is the constant in Lemma 1.6; the necessary quantitative condition on g (which depends on ε and c3) is never stated. Thus both Theorem 1.1 and Theorem 1.2 rest on this unproved input.
  3. [§2, Proposition 3.5] Proposition 3.5, which supplies the minor-arc bound for S4^♯(α;H), is asserted without proof ('By combining the above results, we get the following'). The preceding Proposition 3.4 is an inverse theorem: a large exponential sum forces α to be close to a rational with q ≪ D(δ)^{-1} and ‖qα‖ ≤ q/(HδD(δ)). To obtain a uniform bound on the minor arcs, one must show that every α satisfying these conditions lies in the major arcs M defined in Section 1.2, with Q = X^ε and the given width β(δ). This verification is not included. In addition, the proof of Proposition 3.4 is sketchy and contains notational slips in the Cauchy–Schwarz step and in the use of the conditions D(δ)B ≪ 1 and D(δ)B ≫ 1; a complete proof with correct constants is needed before Theorem 1.1 can be accepted.
  4. [§1.2, Lemma 1.5] The proof of Lemma 1.5 is too compressed and leaves a necessary condition unstated. The bound (1.6) uses the density estimate and the zero-free region to obtain ∑χ |Wi(χ,0)|^2 ≪ H^2 q^{5ε} X^{-1.25c2+o(1)}, and the text then concludes that the left-hand side of (1.4) is bounded by δ^{-3/2} Q logQ H X^{-0.625c2+o(1)}. For this to be o(H), one needs ε to be sufficiently small in terms of c2; the paper only says at the start of Section 1.1 that ε is a sufficiently small constant, but the precise condition is not spelled out. Also, the notation for D(δ) and β(δ) is inconsistent: near (1.4) D(δ) is defined as (log X)^{-36-1} and β(δ) = 1/(HδD(δ)), while the proof writes β(δ) = (log X)^{36+1}/(Hδ). These are the same quantity, but the reader is left to reconstruct the algebra, and the powers of log X in the final bound are not checked. Since Lemma 1.5 is used to discard the cross terms in the major arcs, this section needs a full, self-contained derivation.
minor comments (6)
  1. [§1.3, heading] The heading 'Proof of Theorem 1.3 assuming Theorem 1.2' should read 'Proof of Theorem 1.2'.
  2. [Eq. (1.9) and Theorem 1.1] The set [X,X+H]^* is written as '[X.X +H]∗' in (1.9) and as '[x,x+H]*' in the statement of Theorem 1.1; these typographical inconsistencies should be corrected.
  3. [Proposition 2.5] The statement 'o(|[1.H ]∗|)' should be 'o(|[X,X+H]^*|)'.
  4. [Proposition 2.4] The displayed error term is O(X^{5ε}H^{3ε}), while the text immediately before the display says O(X^{5ε}H^{ε}); the two should be reconciled.
  5. [§1.2, definitions] The exponent in D(δ) := (log X)^{-36-1} would be clearer as -37, and the proof of Lemma 1.5 writes β(δ) = (log X)^{36+1}/(Hδ), which is the same as 1/(HδD(δ)) only after careful parsing.
  6. [§3, Lemma 3.1] In the proof of Lemma 3.1, the use of the pigeonhole principle should be spelled out more fully; as written, the two consecutive applications are hard to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the argument is conditional on an external zero-free-region assumption and imported estimates; the only self-citation is motivational, and the two support gaps are unverified transfers, not circular reductions.

full rationale

The derivation chain for Theorem 1.2 is a standard Hardy-Littlewood circle method, with the missing-digit set entering only as the averaging weight in the minor-arc contribution. The main inputs are the stated zero-free region (Section 1, Assumption), Jutila's zero-density estimate (Lemma 1.3), the standard explicit formula (Lemma 1.4, citing Harman), the external minor-arc estimate delegated to Matomaki-Shao (Lemma 1.6), and the approximant/type-I machinery of Matomaki, Shao, Tao, and Teravainen (Propositions 2.1, 3.4, Lemma 3.3). No fitting of constants to the output occurs, and no displayed quantity is defined in terms of the Goldbach conclusion, so the theorem is not equivalent to its assumptions by construction. The self-citation to the author's earlier paper [9] in the introduction ('In [9], we proved that (1.1) ... We will follow a similar argument') is motivational rather than load-bearing: no estimate used in the proofs is imported from [9]. Two genuine verification gaps are present but are not circularity. First, Lemma 1.6 is dispatched with the sentence 'The proof comes from the argument in [12]. The only difference is, we use delta as a power of X^{-epsilon}', without checking that the hypotheses and constants of [12] survive the substitution; this is an unverified black-box transfer, not a self-referential reduction. Second, Lemma 1.7 is stated without proof or citation and is used crucially in Section 1.3 and Section 2; this is an omitted support, and if it is supplied only by the author's own unpublished work that fact is not shown in the text, but in either case the lemma is a separate Fourier-average input, not the target Goldbach statement. Accordingly, the circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central theorems depend on an assumed zero-free region for Dirichlet L-functions, on the existence of a sufficiently large base g, and on a stockpile of imported results from Jutila, Matomaki-Shao, Matomaki et al., and Dartyge-Mauduit. No constants are fitted to data. Lemma 1.7 is an unproved lemma inside this paper; it is treated as an axiom here because it is load-bearing and the text gives neither proof nor citation. The free parameters are the universal small epsilon and the unspecified zero-free-region constant c2, both quantified existentially rather than data-fitted.

free parameters (2)
  • epsilon = arbitrarily small; no numerical value
    The theorem is quantified over every epsilon>0, but the proof's constants, the condition that g be sufficiently large, and the zero-free-region exponent c2 all implicitly depend on choosing epsilon small enough. It is a hand-chosen parameter, not fitted to data.
  • c2 (zero-free region width) = some fixed positive constant, not specified
    The central assumption fixes an unspecified c2 such that L(s,chi) has no zeros for Re(s)>1-c2. The proof needs c2 large enough relative to epsilon for the o(H) estimates to hold, but no explicit lower bound is given. This is an assumed constant.
assumptions (6)
  • domain assumption Zero-free region for Dirichlet L-functions: L(s,chi) != 0 for Re(s)>1-c2 for all characters chi, for some fixed c2>0.
    Stated in Section 1 and used in Lemma 1.5 (equation (1.6)) and Lemma 1.6 to control prime sums in short intervals.
  • domain assumption The base g is sufficiently large depending on epsilon.
    Theorem 1.1 and Theorem 1.2 require g >= g0(epsilon); without this, the Fourier-average savings in Lemma 1.7 and the minor-arc estimates are not guaranteed. No explicit lower bound is given.
  • standard math Jutila zero-density estimate (Lemma 1.3): N(sigma,T,q) << (qT)^{(2+epsilon)(1-sigma)+o(1)}.
    Imported from [8] and used in the proof of Lemma 1.5.
  • standard math Matomaki-Shao-Tao-Teravainen higher-uniformity and discorrelation results from [11], [12], [13], including the approximation of d4 by d4-sharp (Proposition 2.1) and the inverse theorem (Proposition 3.4).
    The proof of Theorem 1.1 rests on these results; the paper does not reprove them.
  • ad hoc to paper Lemma 1.7: the L1-average of the missing-digit Fourier sum F_{[X,X+H]} is bounded by |[X,X+H]*|^{-1+log((log g)+1)/log(g-1)}.
    Asserted without proof or reference; used in Proposition 2.2 and in the minor-arc estimate of Section 1.3. If it fails, the minor-arc bounds do not yield o(|set|).
  • standard math Dartyge-Mauduit Lemma 2.6 on the distribution of the missing-digit set in arithmetic progressions.
    Imported from [4] to count multiples l in [X,X+H]* and used in the final estimate for Theorem 1.1.

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Cite this review

Pith. "Pith review of Goldbach's Problem in short intervals for numbers with a missing digit." pith.science (2026). https://pith.science/paper/5ZQSES4E

@misc{pith2026241219975,
  author       = {Pith},
  title        = {Pith review of: Goldbach's Problem in short intervals for numbers with a missing digit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ZQSES4E}},
  note         = {Machine review of arXiv:2412.19975}
}
abstract

In this paper, by assuming a zero-free region for Dirichlet L-functions, we show that almost all even integers $n$ in a short interval $[x,x+x^{2/3+\varepsilon}]$ with a missing digit are Goldbach numbers.

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Reference graph

Works this paper leans on

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