{"id":"c170804a-3a49-49a3-a209-7a3a78091cda","arxiv_id":"1908.09439","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves an average upper bound for primes of the form n^4+k, but a suspected error in the dyadic sum in Section 6 undermines the proof.","lead":"This number theory paper claims to estimate how often n^4+k is prime for most k up to x^4. The proof appears to contain a gap in a key estimate, so the main theorem is not established as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 6 dyadic summation drops the y^{1/4} prefactor, so the claimed Ψ2 bound is too small and (6.6) does not follow.","rationale":"The central claim requires the Ψ2 bound (6.6) to be ≪ y(log x)^{-B}. The load-bearing step is the dyadic summation in Section 6. The reader's objection is correct: the y^{1/4} prefactor coming from the square-full part decomposition is omitted when passing from the per-dyadic bound to the summed bound. This is an internal algebraic error, not a disagreement with consensus. The corrected bound is a factor y^{1/4} larger, and with the chosen parameters it is y^{1+ε/4} before the outer Cauchy factor v, so it cannot yield (6.6). I also note the unproved assertion that h(s,k) is absolutely bounded for Re(s) > -1/2 + ε in the Ψ3 section, but the summation error is already decisive. The verdict should remain REJECT because the proof of Theorem 2.1 does not hold as written.","tokens_in":14626,"tokens_out":20179,"duration_ms":197803,"concrete_test":"Recompute the two sums in the displayed 'Summing over r' line of Section 6 with Q_r = 2^{r-1}U and R = floor(log_2(y^{3+ε}/U)): S_1 = y^{1/4-ε'/2} Σ_{r=1}^R (Q_r y)^ε Q_r^{1/4}, S_2 = y^{1/4-ε'/2} Σ_{r=1}^R (Q_r y)^ε y Q_r^{-1/3}. For U = y^{3/4}, verify that S_1 ≍ y^{1+ε/4} and S_2 ≍ y^{5/4}U^{-1/3} up to factors (log y)^{O(1)}, and compare these with the paper's claimed y^{1/4-ε'}(y^{3/4+6ε} + y^{1+ε}U^{-1/3-ε}). If the factors differ by y^{1/4}, then (6.4) and (6.6) fail.","verdict_should_be":"REJECT","load_bearing_attack":"In Section 6, after the quartic large sieve and duality, each dyadic interval Q = 2^{r-1}U is bounded by y^{1/4-ε'/2}(Qy)^ε(Q^{1/4} + y Q^{-1/3}). The paper then sums r = 1,...,R = floor(log_2(y^{3+ε}/U)) and claims y^{1/4-ε'}(y^{3/4+6ε} + y^{1+ε}U^{-1/3-ε}). This summation drops the y^{1/4} prefactor. Indeed, Σ_{r≤R} Q_r^{1/4} ≍ (2^R U)^{1/4}, so the first term is y^{1/4}(2^R U)^{1/4} ≍ y^{1+ε/4}, not y^{3/4+6ε}; similarly Σ y Q_r^{-1/3} ≍ y U^{-1/3}, so the second term is y^{5/4}U^{-1/3}, not y^{1+ε}U^{-1/3-ε}. Thus the displayed line is smaller by a factor y^{1/4}. With the paper's choices U = y^{3/4} and ε = ε'/28, the corrected dyadic sum is ≫ y^{1+ε/4} before the outer v-factor in (6.4), which cannot be ≤ y(log x)^{-B}. Hence the Ψ2 estimate (6.4) and the final claim (6.6) are unsupported, and Theorem 2.1 does not follow from the argument as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a quartic analogue of the Baier--Zhao quadratic average result: for fixed A, B, epsilon > 0 and x^4 (log x)^{-A} <= y <= x^4, the second moment over k <= y with square-full part <= y^{1/2-epsilon} of sum_{n<=x} Lambda(n^4+k) - S(k)x is O(y x^2 (log x)^{-B}), with S(k) the Euler product over primes p>2 of (1 - (n_{k,p}-1)/(p-1)). The proof uses the circle method: Sections 4--7 treat the major arc and Section 8 the minor arc; the main technical difficulty is the tail Psi(k) of the singular series, split into Psi_1, Psi_2, Psi_3 and bounded with a large sieve for quartic characters, duality, Perron's formula, and Hecke L-functions. I note at the outset that the dyadic-summation concern in the stress-test does not land: the displayed bound after the r-sum already contains the prefactor y^{1/4-epsilon'}, so summing Q^{1/4} to (2^R U)^{1/4} gives y^{1+O(epsilon)} as written. The actual problem is elsewhere: the singular series is incorrectly reduced to primes congruent to 1 mod 4.","tokens_in":14904,"tokens_out":31436,"duration_ms":285868,"significance":"If Theorem 2.1 were correct, it would be a natural extension of the quadratic and cubic results of Baier--Zhao and Foo--Zhao to quartic polynomials, and the Corollary giving almost-all k would be a useful contribution. The paper has genuine strengths: no fitted parameters, a direct analytic proof against the conjectured singular series, and a substantive use of the quartic large sieve, duality, and Hecke L-functions. Unfortunately, the proof as written contains a false statement about the singular series at primes p ≡ 3 mod 4, and this invalidates the main term and the tail estimate on which Theorem 2.1 rests.","major_comments":[{"comment":"The displayed claim \"Sigma(p)=0 if p=2 or p≡3 mod4\" contradicts the immediately preceding identity Sigma(p)=p(n_{k,p}-1). For p≡3 mod4 and p∤k, the congruence m^4+k≡0 mod p has either 0 or 2 solutions, so Sigma(p) is -p or p, not 0. For example, p=3 and k=1 give n_{1,3}=0 and hence Sigma(3)=-3. Consequently the subsequent statement \"we can restrict q to be square-free with prime factors that are congruent to 1 modulo 4\" is false, and the tail Psi(k) defined in (5.1) omits all contributions from primes p≡3 mod4. Section 6 estimates only the p≡1 part via quartic characters and the Hecke L-function associated with f(s,k), so the bound (6.6) does not control the full tail Psi(k). Theorem 2.1 is therefore not established by the argument as written.","section":"Section 5, formula for Sigma(p)"},{"comment":"The statement \"It can be shown that h is absolutely bounded for all Re(s)>-1/2+epsilon\" is asserted without proof or citation. This bound is needed for the Perron inversion leading to (6.5); the Dirichlet series for L(s+1,(k/.)_4) is not absolutely convergent on that half-plane, so the cancellation encoded in h is not automatic. As written, the estimate (6.5) is unsupported.","section":"Section 6, treatment of Psi_3"}],"minor_comments":[{"comment":"The title contains a typo: \"ON A VERAGE\" should read \"ON AVERAGE\".","section":"Title"},{"comment":"There are several typographical errors, including \"speacial\" for \"special\", \"weill-known\" for \"well-known\", \"non-principle\" for \"non-principal\", and \"von-Mongoldt\" for \"von Mangoldt\".","section":"Throughout"},{"comment":"Theorem 2.1 uses a parameter epsilon in the condition kappa(k) <= y^{1/2-epsilon}, while the proof works with epsilon' and sets epsilon = epsilon'/28 only near the end of Section 6. The relation between the two parameters should be stated in the theorem or at the beginning of the proof.","section":"Theorem 2.1 and Section 6"},{"comment":"The notation \"v = log 2(exp(y^{epsilon/3})/U)\" is ambiguous; clarify that this denotes the base-2 logarithm of the quotient, and state the resulting size of v, since it is multiplied by the whole bound in (6.4).","section":"Section 6, parameter v"},{"comment":"In (6.3) the notation kappa(n) is used for Gaussian integers n in Z[i], but kappa was defined in Section 2 only for rational integers; a definition or explanation should be added.","section":"Section 6, equation (6.3)"}],"recommendation":"reject","confidential_remarks":"I agree with the rejection recommendation, but for a different reason from the attached stress-test: the claimed missing y^{1/4} factor in Section 6 does not survive a careful reading, since the prefactor y^{1/4-epsilon'} is present in the displayed bound. The actual obstruction is the incorrect treatment of primes p≡3 mod4 in the singular series, which is a load-bearing error in the main term and in the definition of Psi(k)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real attempt to push the Baier–Zhao and Foo–Zhao circle-method from r=2,3 to r=4, using the quartic character large sieve and quartic reciprocity. The major arc singular series computation and the minor arc Weyl-type bound are competently handled, and the paper is clearly written. The case r=4 is genuinely new, and the machinery is appropriate for it.\n\nThe problem is in Section 6. After the large sieve and duality, the bound for each dyadic interval Q ~ 2^r U is y^{1/4-ε'/2}(Qy)^ε(Q^{1/4}+yQ^{-1/3}). The paper then sums this over r=1,...,R with 2^R U = y^{3+ε} and claims y^{1/4-ε'}(y^{3/4+6ε}+y^{1+ε}U^{-1/3-ε}). That does not follow. The first term summed is roughly y^{1/4-ε'/2}·(2^R U)^{1/4} ≈ y^{1-ε'/2+O(ε)}, not y^{3/4+6ε}; the y^{1/4} prefactor appears to have been dropped. With the chosen U=y^{3/4}, the corrected Ψ2 contribution is ≫ y^{1+ε/4}, which cannot be made ≤ y(log x)^{-B}. So (6.4) and (6.6) do not follow, and Theorem 2.1 is unsupported by the argument as written.\n\nA second, smaller soft spot: the Ψ3 estimate relies on an assertion that h(s,k) is absolutely bounded for Re(s)>-1/2+ε, but this is stated without proof. It may be true, but it is load-bearing and needs a reference or a short argument.\n\nIf the dyadic summation is fixed, the strategy is plausible and this could become a correct paper. As it stands, the proof has a concrete gap in a central estimate. I would still send it to a referee with a note to check Section 6 carefully; the error is identifiable and the rest deserves scrutiny. But I would not accept it as is.","headline":"A serious quartic extension of the Baier–Zhao/Foo–Zhao method, but a dyadic summation slip in Section 6 drops a y^{1/4} factor and the claimed Ψ2 bound does not follow.","tokens_in":15436,"tokens_out":3717,"would_cite":false,"duration_ms":32476,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P55","11L07","11N13","11N32"],"pacs":[],"model":"deepseek-v4-flash","headline":"For almost every constant k with small square-full part, n^4+k takes prime values exactly as the Bateman-Horn singular series predicts, on average over k.","keywords":["circle method","quartic characters","primes in polynomial progressions","singular series","large sieve","square-full part","second moment","Bateman-Horn conjecture"],"falsifier":"Recompute the Section 6 dyadic sum: as Q ranges over powers of two from U up to $y^{{3+ε}}$, the contribution $y^{{1/4}}$$Q^{{1/4}}$ accumulates to about y, while the paper's display claims $y^{{3/4+6ε}}$; carrying out this summation settles whether the Ψ_2 bound and hence Theorem 2.1 follow.","tokens_in":14354,"feed_emoji":"🔢","tokens_out":10280,"duration_ms":95013,"temperature":0.7,"pith_summary":"The paper proves an average (second-moment) version of the Bateman-Horn conjecture for quartic polynomials of the form $n^{4}$+k. It shows that when one averages over constants k up to y (with $x^{4}$(log x)^{-A} ≤ y ≤ $x^{4}$) and restricts to k whose square-full part (the square factor $ℓ^{2}$ in k=$ℓ^{2}$ m with m squarefree) is not too large, the count of primes $n^{4}$+k with n≤x is S(k)x plus an error that is small on average. The error term is strong enough to imply that for almost all such k, the expected number of primes holds up to O(x/(log x)^B). This extends to degree four the average results previously known for quadratic and cubic progressions.","feed_headline":"Quartic primes appear on schedule for almost all shifts k","feed_subtitle":"The Bateman-Horn prediction for n^4+k is confirmed on average over k up to x^4.","key_machinery":"The argument is a circle-method decomposition of the exponential sum over $n^{4}$, with the singular series S(k) emerging from complete character sums modulo q. The discrepancy from S(k)x is split into three tail sums Ψ_1, Ψ_2, Ψ_3 over ranges of moduli. Ψ_1 is handled elementarily, Ψ_2 uses a large sieve inequality for quartic characters together with the duality principle, and Ψ_3 is estimated through Perron's formula and Hecke L-functions with a zero-free region. The major-arc error terms are controlled by Gallagher's large sieve and Mikawa's estimate for character sums over primes.","core_discovery":"Theorem 2.1 establishes the second-moment bound ∑_{k≤y, κ(k)≤$y^{{1/2-ε}}$} |∑_{n≤x} Λ($n^{4}$+k) - S(k)x|^2 ≪ $yx^{2}$/(log x)^B, for any fixed A,B,ε>0 and $x^{4}$(log x)^{-A} ≤ y ≤ $x^{4}$, where κ(k) is the square-full part of k, Λ is the von Mangoldt function, and S(k) = ∏_{p>2} (1 - (n_p-1)/(p-1)) with n_p the number of solutions to $n^{4}$+k≡0 mod p. The corollary is that for almost all such k, ∑_{n≤x} Λ($n^{4}$+k) = S(k)x + O(x/(log x)^B). This is the quartic analogue of the Baier-Zhao quadratic result and the Foo-Zhao cubic result.","pith_inferences":["Editorial extension: The same second-moment strategy should adapt to polynomials n^r+k for r≥5 if a large sieve for r-th order characters with comparable exponents is available; the quartic sieve exponents (Q^{5/4}+Q^{2/3}M) are the natural bottleneck.","Editorial extension: The square-full restriction suggests an open problem the paper does not resolve: whether the second moment can be extended to all k≤y by treating conductors with large square factors through a different sieve, without changing the singular-series main term.","Editorial extension: A reader who wants to test the robustness of the proof should isolate the dyadic summation in Section 6; verifying that summation is the fastest way to decide whether the stated Ψ_2 bound follows from the large sieve estimate."],"forward_implications":["For almost all k with square-full part at most y^{1/2-ε}, the expected Bateman-Horn count for n^4+k holds to within x/(log x)^B.","The admissible range of k, from x^4(log x)^{-A} up to x^4, matches in shape the ranges proved in the quadratic and cubic cases.","The exception set for the corollary has size O(y/(log x)^C), so the failure of the pointwise prediction is rare in a quantitative sense.","The restriction to small square-full part is explicit in the theorem, so any attempt to cover all k must overcome the large-square-full regime."],"supporting_citations":[{"why":"It supplies the circle-method template and singular-series formulation that the quartic proof adapts from the quadratic mean-value result.","marker":"[1]"},{"why":"It states the heuristic singular-series prediction that the theorem verifies on average for n^4+k.","marker":"[4]"},{"why":"It is the immediate cubic precedent whose major-arc error term treatment is transferred to the quartic setting.","marker":"[6]"},{"why":"It is the large sieve density estimate used to bound the major-arc error terms.","marker":"[8]"},{"why":"It provides the large sieve inequality for quartic characters, the main tool in the Ψ_2 estimate.","marker":"[9]"},{"why":"It supplies the large sieve for algebraic number fields that, with quartic reciprocity, controls the large-modulus tail Ψ_3.","marker":"[14]"},{"why":"It provides the Pólya-Vinogradov and Weyl-shift bounds used in the major- and minor-arc estimates.","marker":"[17]"},{"why":"It gives the estimate for character sums over primes that controls the J(q,Δ) terms in the major-arc error analysis.","marker":"[20]"}],"fun_headline_variants":["Quartic primes: expected count for almost all shifts k","Average result: n^4+k has the expected primes for most k","Bateman-Horn for quartic polynomials holds on average","For most k, primes of form n^4+k match the predicted density","For shifts with small square-full part, quartic prime counts match prediction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof hinges on one summation step in Section 6: adding the quartic large-sieve estimates over exponentially growing ranges of moduli, whose total size the paper estimates in a single displayed bound.","fun_headline_variants_meta":{"raw":{"variants":["Quartic primes: expected count for almost all shifts k","Average result: n^4+k has the expected primes for most k","Bateman-Horn for quartic polynomials holds on average","For most k, primes of form n^4+k match the predicted density","For shifts with small square-full part, quartic prime counts match prediction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000546,"raw_usage":{"total_tokens":2530,"prompt_tokens":785,"completion_tokens":1745,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":401,"completion_tokens_details":{"reasoning_tokens":1654}},"tokens_in":401,"tokens_out":1745,"duration_ms":15018,"temperature":1.0,"reasoning_tokens":1654,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:12:35.467730+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the Section 6 dyadic sum: as Q ranges over powers of two from U up to $y^{{3+ε}}$, the contribution $y^{{1/4}}$$Q^{{1/4}}$ accumulates to about y, while the paper's display claims $y^{{3/4+6ε}}$; carrying out this summation settles whether the Ψ_2 bound and hence Theorem 2.1 follow.","supporting_citations":[{"cited_title":"Baier, L","cited_arxiv_id":null,"evidence_quote":"It supplies the circle-method template and singular-series formulation that the quartic proof adapts from the quadratic mean-value result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It states the heuristic singular-series prediction that the theorem verifies on average for n^4+k."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the immediate cubic precedent whose major-arc error term treatment is transferred to the quartic setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the large sieve density estimate used to bound the major-arc error terms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the large sieve inequality for quartic characters, the main tool in the Ψ_2 estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the large sieve for algebraic number fields that, with quartic reciprocity, controls the large-modulus tail Ψ_3."},{"cited_title":"Iwaniec, E","cited_arxiv_id":null,"evidence_quote":"It provides the Pólya-Vinogradov and Weyl-shift bounds used in the major- and minor-arc estimates."},{"cited_title":"Mikawa, On prime twins","cited_arxiv_id":null,"evidence_quote":"It gives the estimate for character sums over primes that controls the J(q,Δ) terms in the major-arc error analysis."}],"review_version":1}