{"id":"9e2778d4-fdcd-418a-8cff-c92f6d67f97c","arxiv_id":"2412.19975","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Assuming a zero-free region for Dirichlet L-functions, almost all even integers with a missing digit in intervals of length x^{2/3+epsilon} are sums of two primes.","lead":"This paper proves a conditional result: if Dirichlet L-functions have no zeros very close to the 1-line, then almost all large even numbers that avoid one digit in their base-g expansion are sums of two primes, even inside short intervals. The proof adds a new divisor-function estimate over such digit-restricted numbers.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 1.6 is the load-bearing transfer: a one-sentence delegation to Matomäki–Shao with δ=X^{-ε}, without verifying that H≥X^{2/3+ε} and the Farey minor-arc width β(δ) satisfy the cited theorem's hypotheses.","rationale":"Read in good faith, the paper proposes a plausible conditional circle-method proof. Major arcs are handled under the zero-free-region assumption, and Theorem 1.1 gives the required d4 bound. The strongest dependence is the minor-arc estimate for primes in short intervals, because without it the sum over the missing-digit set on [0,1] cannot be shown to be o(H|[X,X+H]^*|). Lemma 1.7 is also unproved, but it is a Fourier-average estimate for the missing-digit set that is similar to (and weaker than) known bounds from Maynard's work, so it is less likely to be the decisive failure point. Lemma 1.6, by contrast, imports a deep theorem from [12]/[11] with only a sentence and no check of the H-range or constants; the whole theorem is conditional on that transfer. The reader's verdict CONDITIONAL is therefore appropriate. My read does not change it; it sharpens the condition: the gap is Lemma 1.6's unverified transfer, which should be closed before the claim is regarded as proven. No evidence of circularity or fabrication was found; the concern is about an under-specified citation.","tokens_in":10913,"tokens_out":22140,"duration_ms":227004,"concrete_test":"Locate the exact theorem in Matomäki–Shao [12] (and [11, Lemma 3.4]) from which Lemma 1.6 is supposed to follow. Write its hypotheses with all constants; substitute δ=X^{-ε}, H=X^{2/3+ε}, and the paper's definition of m. Check (i) the allowed range of H, (ii) the relation between Q and the major-arc width, and (iii) whether the constant c3 is uniform and positive after the substitution. If the cited theorem requires, for example, H≥X^{5/6+ε} or a smaller Q, then Lemma 1.6 is false in the needed range and Theorem 1.2 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 1.2 depends on the minor-arc bound Lemma 1.6 (Section 1.2): for α outside the major arcs M, max(|S1(α)|,|S2(α)|) ≪ H X^{-c3 ε}. Its proof is a single sentence citing [12] and saying the only difference is taking δ=X^{-ε}, possibly using [11, Lemma 3.4]. This is not a verification. The minor arcs here are defined with Q=δ^{-1}=X^ε and β(δ)=(log X)^{37}/(Hδ); the cited results have their own constants and ranges, and nothing in the text shows that the substitution δ=X^{-ε} preserves the condition H≥X^{2/3+ε}, nor that c3 can be chosen independent of g, nor that the quantitative input from [11, Lemma 3.4] is compatible with the very narrow β(δ). If the true range in [12] is H≥X^{5/6+ε}, or if Q must be a smaller power of X, Lemma 1.6 fails and the minor arcs contribute too much. The final comparison also silently needs g so large that log(log g+1)/log(g-1) is below the power saving 2c3ε; this dependence is not stated. Thus the theorem rests on an unverified black-box transfer.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11238,"tokens_out":10131,"duration_ms":90964,"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":[{"comment":"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.","section":"§1.2, Lemma 1.6"},{"comment":"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.","section":"§1.2, Lemma 1.7"},{"comment":"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.","section":"§2, Proposition 3.5"},{"comment":"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.","section":"§1.2, Lemma 1.5"}],"minor_comments":[{"comment":"The heading 'Proof of Theorem 1.3 assuming Theorem 1.2' should read 'Proof of Theorem 1.2'.","section":"§1.3, heading"},{"comment":"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.","section":"Eq. (1.9) and Theorem 1.1"},{"comment":"The statement 'o(|[1.H ]∗|)' should be 'o(|[X,X+H]^*|)'.","section":"Proposition 2.5"},{"comment":"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.","section":"Proposition 2.4"},{"comment":"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.","section":"§1.2, definitions"},{"comment":"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.","section":"§3, Lemma 3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper has a promising outline and the main theorems are likely correct, but as submitted it is not sufficiently self-contained: Lemma 1.6 and Lemma 1.7 are load-bearing and unproved, and Proposition 3.5 is stated without proof. I recommend a major revision that asks for complete proofs of these three items, with an explicit verification of the Matomäki–Shao transfer. The editor may also wish to check whether the zero-free-region assumption is strong enough and stated consistently with the applications in [12]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result in the making, not a finished proof. The new thing is the combination of Perelli-Pintz style short-interval Goldbach with the missing-digit set, and the d4 divisor moment in Theorem 1.1. The strategy is sensible and the major-arc handling is standard. The stress-test worry about Lemma 1.6 is right: the central minor-arc bound for prime sums is delegated to Matomäki-Shao in one sentence, with δ=X^{-ε}, and the text never checks that H≥X^{2/3+ε} and the Farey arc width β(δ) satisfy the cited theorem's hypotheses. That is load-bearing. If the transfer fails, the minor arcs swamp the main term. Lemma 1.7, the L1 average of the missing-digit exponential sum, is stated without proof and is used twice; it needs either a proof or a precise citation, not an assertion. The final comparison also silently needs g huge enough to beat the X^{-cε} saving with a log(log g+1)/log(g-1) factor; that dependence is not quantified. These are fillable gaps for an expert, but they are real gaps. To the author's credit, there is no fitting or circularity; the zero-free-region assumption is an external hypothesis, stated clearly, and the cited heavy tools are legitimate. The self-citations are fine. If the gaps close, this is a meaningful extension of Cumberbatch's long-interval result. As written, I would send it to a referee, with instructions to check Lemma 1.6 and Lemma 1.7 first. The author should either fully prove the transfer or restate the cited theorems with all parameters and ranges. The idea is good and the skeleton is there; peer review can fix this. Verdict: conditional acceptance, and I would not cite it as a theorem until the black boxes are unpacked.","headline":"New short-interval Goldbach result for missing-digit numbers, but the core minor-arc bound is a one-line citation transfer that needs verification.","tokens_in":11764,"tokens_out":1798,"would_cite":false,"duration_ms":17236,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P32","11N05","11N37","11A63"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Goldbach numbers","missing digits","short intervals","Dirichlet L-functions","zero-free region","divisor function","circle method","exponential sums"],"falsifier":"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.","tokens_in":10664,"feed_emoji":"🔢","tokens_out":18443,"duration_ms":163304,"temperature":0.7,"pith_summary":"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.","feed_headline":"Even digit-missing numbers are almost all Goldbach sums","feed_subtitle":"Under a zero-free region, short intervals of even numbers with a banned digit are almost all sums of two primes.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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"],"forward_implications":["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"],"supporting_citations":[{"why":"Supplies the short-interval circle-method setup and the exceptional-set estimate that the paper adapts to missing-digit sets.","marker":"[16]"},{"why":"Supplies the minor-arc bound on the weighted prime exponential sums (Lemma 1.6), the main external input.","marker":"[12]"},{"why":"Provides the d4 approximation and the type-I sum framework used to prove Theorem 1.1.","marker":"[13]"},{"why":"Supplies the smoothed decomposition of d4^# into type-I sums used in the minor-arc estimate.","marker":"[11]"},{"why":"Gives the zero-density estimate for Dirichlet L-functions used to control the major-arc error terms.","marker":"[8]"},{"why":"Supplies the standard explicit formula for sums of Λ(n)χ(n), which yields the major-arc approximation (Lemma 1.4).","marker":"[7]"},{"why":"Gives the equidistribution of missing-digit numbers in arithmetic progressions used in Lemma 2.6 to control divisor-sum averages.","marker":"[4]"}],"fun_headline_variants":["Under zero-free region, most digit-missing evens are Goldbach sums","Zero-free region: almost all missing-digit evens are Goldbach","Assuming zero-free region, almost all digit-missing evens are Goldbach","Short intervals + zero-free region: missing-digit evens nearly all Goldbach","Most evens with a banned digit are Goldbach sums under zero-free region"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Under zero-free region, most digit-missing evens are Goldbach sums","Zero-free region: almost all missing-digit evens are Goldbach","Assuming zero-free region, almost all digit-missing evens are Goldbach","Short intervals + zero-free region: missing-digit evens nearly all Goldbach","Most evens with a banned digit are Goldbach sums under zero-free region"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001403,"raw_usage":{"total_tokens":5643,"prompt_tokens":889,"completion_tokens":4754,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":505,"completion_tokens_details":{"reasoning_tokens":4665}},"tokens_in":505,"tokens_out":4754,"duration_ms":37536,"temperature":1.0,"reasoning_tokens":4665,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:45:14.576457+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Perelli and J","cited_arxiv_id":null,"evidence_quote":"Supplies the short-interval circle-method setup and the exceptional-set estimate that the paper adapts to missing-digit sets."},{"cited_title":"Discorrelation bet ween primes in short intervals and polynomial phases","cited_arxiv_id":null,"evidence_quote":"Supplies the minor-arc bound on the weighted prime exponential sums (Lemma 1.6), the main external input."},{"cited_title":"Higher uniformity of arithmetic functions in short intervals I","cited_arxiv_id":null,"evidence_quote":"Provides the d4 approximation and the type-I sum framework used to prove Theorem 1.1."},{"cited_title":"Higher uniformity of arithmetic functions in short interva ls ii","cited_arxiv_id":null,"evidence_quote":"Supplies the smoothed decomposition of d4^# into type-I sums used in the minor-arc estimate."},{"cited_title":"On Linnik’s constant","cited_arxiv_id":null,"evidence_quote":"Gives the zero-density estimate for Dirichlet L-functions used to control the major-arc error terms."},{"cited_title":"Prime-detecting sieves, volume 33 of London Mathematical Society Monographs Series","cited_arxiv_id":null,"evidence_quote":"Supplies the standard explicit formula for sums of Λ(n)χ(n), which yields the major-arc approximation (Lemma 1.4)."},{"cited_title":"Ensembles de den sitĂŠ nulle contenant des entiers possĂŠdant au plus deux facteurs premiers","cited_arxiv_id":null,"evidence_quote":"Gives the equidistribution of missing-digit numbers in arithmetic progressions used in Lemma 2.6 to control divisor-sum averages."}],"review_version":1}