Pith. sign in

REVIEW 2 major objections 5 minor 20 references

On primes represented by quartic polynomials on average

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.09439 v1 pith:UJIDACBD submitted 2019-08-26 math.NT

Kam Hung Yau This is my paper · ORCID
classification math.NT MSC 11P5511L0711N1311N32
keywords circlemethodquarticcharactersprimesinpolynomialprogressionssingularserieslargesievesquare-fullpartsecondmomentBateman-Hornconjecture
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Section 5, formula for Sigma(p)] 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.
  2. [Section 6, treatment of Psi_3] 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.
minor comments (5)
  1. [Title] The title contains a typo: "ON A VERAGE" should read "ON AVERAGE".
  2. [Throughout] 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".
  3. [Theorem 2.1 and Section 6] 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.
  4. [Section 6, parameter v] 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).
  5. [Section 6, equation (6.3)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theorem is an independent analytic estimate against external lemmas.

full rationale

The paper does not fit any parameter to the quantity it predicts. The singular series S(k) is defined by an Euler product over primes (1 - (n_p - 1)/(p - 1)), and the main term S(k)x is derived from the major arc by standard character-sum orthogonality and Ramanujan-sum evaluation; it is not defined so as to force Theorem 2.1. The second-moment bounds for Psi_1, Psi_2, and Psi_3 are proved using external results: the quartic large sieve of Gao and Zhao, the large sieve for number fields of Huxley, Pólya-Vinogradov, Gallagher's lemma, Perron's formula, Mikawa's result, and the Weyl bound. None of these inputs contain the target second-moment bound as an assumption, and the paper does not import any uniqueness or structural theorem from its own authors. The only potentially problematic passage, the dyadic summation in Section 6, concerns the size of an error term; if the summation is wrong, that is a correctness defect, not a circular reduction. Since the claimed derivation is self-contained and directly estimates the stated averages against external, parameter-free theorems, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof relies on standard analytic number theory tools: large sieve inequalities, zero-free regions, Perron's formula, and quartic reciprocity. The only ad hoc input is the asserted boundedness of h(s,k), which is not proved in the text. No free parameters are fitted; the singular series is an explicit Euler product.

assumptions (5)
  • standard math Large sieve inequality for quartic characters (Lemma 3.1) imported from Gao and Zhao.
    Used in Section 6 to bound Psi_2. The exact form with Q^{5/4} + Q^{2/3} M is crucial for the argument.
  • standard math Large sieve inequality for number fields (Lemma 3.3) imported from Huxley.
    Used in Section 6 for large conductors, after mapping quartic characters to residues in Z[i].
  • standard math Zero-free region for Hecke L-functions (Lemma 3.10) imported from Iwaniec-Kowalski.
    Used to obtain the exponential decay in the Psi_3 bound via Perron's formula.
  • ad hoc to paper The function h(s,k) is absolutely bounded for all Re(s) > -1/2 + epsilon.
    Stated without proof in Section 6 after defining h(s,k). The Perron integral estimate for Psi_3 depends on this bound.
  • standard math Quartic reciprocity for residue symbols in Z[i].
    Invoked to transform quartic characters into cubic residue symbols, needed for applying the number field large sieve.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On primes represented by quartic polynomials on average." pith.science (2026). https://pith.science/paper/UJIDACBD

@misc{pith2026190809439,
  author       = {Pith},
  title        = {Pith review of: On primes represented by quartic polynomials on average},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UJIDACBD}},
  note         = {Machine review of arXiv:1908.09439}
}
abstract

We obtain an upper bound for the distribution of primes in the form $n^4 + k$ up to $x$, averaged over $k$ with small square-full part. As a corollary, we show that for almost all $k$, there is an expected amount of primes in the form $n^4 +k$ up to $x$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Baier, L

    S. Baier, L. Zhao, Primes in quadratic progressions on average. Math. Ann. 338 (4) (2007), 963–982. 2

    work page 2007

  2. [2]

    Baier, L

    S. Baier, L. Zhao, On primes represented by quadratic polynomials. Anatomy of integers, 159–166, CRM Proc. Lecture Notes 46, Amer. Math. Soc., 2008. 2

    work page 2008

  3. [3]

    Baier, L

    S. Baier, L. Zhao, On primes in quadratric progressions. Int. J. Number Theory 5 (6) (2009), 1017-1035. 2

    work page 2009

  4. [4]

    P. T. Bateman, R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers. Math. Comp. 16, 1962, 363–367. 1

    work page 1962

  5. [5]

    Davenport, Multiplicative number theory

    H. Davenport, Multiplicative number theory. Third edition. Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000. 4

    work page 2000

  6. [6]

    T. Foo, L. Zhao, On primes represented by cubic polynomials. Math. Z. 274 (1-2) (2013), 323–340. 2, 9, 15, 18

    work page 2013

  7. [7]

    J. B. Friedlander, H. Iwaniec, The polynomial X 2 + Y 4 captures its primes. Ann. of Math. (2) 148 (3) (1998), 945–1040. 1

    work page 1998

  8. [8]

    P. X. Gallagher, A large sieve density estimate near σ = 1. Invent. Math. 11, 1970, 329–339. 5

    work page 1970

Show all 20 references
  1. [9]

    P. Gao, L. Zhao, Large sieve inequalities for quartic characters. Q. J. Math. 63 (4) (2012), 891–917. 3, 13 PRIMES REPRESENTED BY QUARTIC POLYNOMIALS ON A VERAGE 21

    2012

  2. [10]

    P. R. Halmos, Finite-Dimensional Vector Spaces. D. Van Nostrand, New York (1958). 3, 5

    1958

  3. [11]

    G. H. Hardy, J. E. Littlewood, G. P´ olya, Inequalities. Cambridge University Press, Cambridge (1964). 4

    1964

  4. [12]

    D. R. Heath-Brown, Primes represented by x3 + 2 y3. Acta Math. 186 (1) (2001), 1–84. 1

    2001

  5. [13]

    D. R. Heath-Brown, X. Li, Prime values of a2 + p4. Invent. Math. 208 (2) (2017), 441–499. 1

    2017

  6. [14]

    M. N. Huxley, The large sieve inequality for algebraic number fields. II. Means of moments of Hecke zeta-functions. Proc. London Math. Soc. ( 3) 21 (1970), 108–128. 3, 5

    1970

  7. [15]

    Ireland, M

    K. Ireland, M. Rosen, A classical introduction to modern number theory. Sec- ond edition. Graduate Texts in Mathematics 84. Springer-Verlag, New York,

  8. [16]

    Iwaniec, The half dimensional sieve

    H. Iwaniec, The half dimensional sieve. Acta Arith. 29 (1) (1976), 65–95. 1

    1976

  9. [17]

    Iwaniec, E

    H. Iwaniec, E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications 53. American Mathematical Society, Prov- idence, RI, 2004. 3, 4, 5

    2004

  10. [18]

    Kuhn, ¨Uber die Primteiler eines Polynomsw

    P. Kuhn, ¨Uber die Primteiler eines Polynomsw. Proceedings of the Interna- tional Congress of Mathematicians. Vol. 2 (Amsterdam, 1954), Er ven P. No- ordhoff N. V., Groningen; North-Holland, Amsterdam, 1954, 35–37 . 1

    1954

  11. [19]

    Ju. V. Linnik, The dispersion method in binary additive problems. American Mathematical Society, Providence, R.I. 2

  12. [20]

    Mikawa, On prime twins

    H. Mikawa, On prime twins. Tsukuba J. Math. 15 (1) (1991), 19–29. 5 Department of Pure Mathematics, University of New South W al es, Sydney, NSW 2052, Australia E-mail address : kamhung.yau@unsw.edu.au

    1991

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.