REVIEW 5 major objections 4 minor 7 references
The average number of representations of an integer as a sum of two prime powers over multiples of a fixed integer
T0 review · 5 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash
Pith's one-line read For fixed q and k, the average number of representations of an integer as a sum of two prime k-th powers, taken over multiples of q, is a constant multiple of the unrestricted average — and that constant can be zero.
desk verdict Genuinely new generalization with a nice character sum, but the I2 error estimate has two concrete gaps that leave Theorem 1.3 unproven as written. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The proof decomposes the generating function F_{q,k}(z) into Dirichlet characters, reducing the problem to exponential sums Ψ_k(z, χ) = Σ_n χ(n)Λ(n) z^{n^k}. An explicit formula, assuming GRH, expresses Ψ_k(z, χ) in terms of a sum over non-trivial zeros of L(s, χ). The main term is isolated as the contribution of characters satisfying χ^k = χ_0, producing the character sum Σ_k(q). The error term is controlled using Gallagher's lemma applied to short intervals of n, together with estimates for the mean-square of the summatory function ψ_k(x, χ).
What would settle it
Compute I_2(N,h) directly for a small case, say k = 2, q = 3, and a non-principal character χ with χ^2 ≠ χ_0, using the standard Gallagher lemma on the sequence n^2 instead of n, and compare the resulting bound with the claim I_2 ≪ N^{1/k} log^3 N. A divergence would indicate that the short-interval adaptation in Section 3.3 is invalid.
Extended reading notes
Core claim
Theorem 1.3: For fixed q ≥ 2 and k ≥ 2, under GRH for Dirichlet L-functions modulo q, the average G_{q,k}(N) satisfies G_{q,k}(N) = Σ_k(q)/φ(q) · G_{1,k}(N) + O(N^{1/k} log^2 N log q / φ(q)), where Σ_k(q) = Σ_{χ^k = χ_0} χ(−1). The main term is therefore a constant multiple of the unrestricted average G_{1,k}(N), and the constant is an explicit character sum that can vanish. When Σ_k(q) = 0, the average over multiples of q is of smaller order than the unrestricted average, illustrating that k-th powers are not uniformly distributed among residue classes modulo q.
Load-bearing premise
The proof assumes that Gallagher's lemma, applied in Section 3.3 to the exponential sum whose phase is n^k α, may be written with short intervals over n (x < n ≤ x+h) rather than over the frequencies n^k; the paper gives no proof of this adaptation, and if it fails the bound for I_2 collapses.
Editorial extensions
If this is right
- If the theorem is correct, the average number of representations over multiples of q is exactly proportional to the unrestricted average, with proportionality factor Σ_k(q)/φ(q).
- For cases where Σ_k(q) = 0 (e.g., q = 3, k = 2), the main term vanishes, so the average over multiples of q is asymptotically smaller than the unrestricted average.
- The result extends the k = 1 theorem of Ikeda and Suriajaya to all k ≥ 2, with a refined error term that depends on q only through log q / φ(q).
- The evaluation of Σ_k(q) gives a complete multiplicative formula, showing exactly when the constant is zero, one, or φ(p^α)-related, depending on the greatest common divisor of k and φ(p^α).
Reading between the lines
- A testable extension is to consider j ≥ 3 prime k-th powers; the paper mentions such work in progress, but the same character-sum mechanism would likely produce a constant multiple involving a higher-order character sum.
- The vanishing of Σ_k(q) suggests that the equation m_1^k + m_2^k ≡ 0 mod q with (m_1 m_2, q) = 1 has no solutions when Σ_k(q) = 0; counting such solutions directly for small q and k would provide a concrete check of Lemma 4.1.
- One could ask whether the error term O(N^{1/k} log^2 N log q / φ(q)) is sharp; if a sharper lower bound for I_2 could be proved without the adapted Gallagher step, it would confirm or refute the validity of that adaptation.
- The proportionality constant being possibly zero may have consequences for sieve or circle-method treatments of prime powers in arithmetic progressions: the expected main term can be rescued only by characters with χ^k = χ_0.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims Theorem 1.3: for fixed q≥2 and k≥2, assuming GRH for modulus q, the average G_{q,k}(N) of the number of representations of n≤N, q|n, as a sum of two prime k-th powers equals (Σ_k(q)/φ(q)) G_{1,k}(N) plus an error term O(N^{1/k} log^2 N log q / φ(q)). The proof uses a Dirichlet character decomposition of the generating function, an explicit formula for Ψ_k(z,χ) under GRH, estimates for character sums in short intervals, and an explicit evaluation of the constant Σ_k(q). This generalizes the k=1 result of Ikeda and Suriajaya.
Significance. If the result holds, it is a natural and worthwhile generalization of the Goldbach-type average over multiples of q to prime powers, and it uncovers a new phenomenon: the proportionality constant Σ_k(q) can vanish, so the restricted average may be of lower order than the unrestricted one. The explicit evaluation of Σ_k(q) in Lemma 4.1 is a concrete, self-contained contribution. However, the proof as written contains several load-bearing gaps, especially in the short-interval estimates of Section 3.3–3.5. These make the current version unsuitable for publication until the technical issues are resolved.
major comments (5)
- [§3.3] The proof applies Montgomery's Gallagher lemma (Lemma 1.9 of [7]) to Ψ_k(z,χ)=∑ χ^k(n)Λ(n)e^{-n^k/N}e(n^kα), whose frequencies are n^k. The standard lemma applies to ∑ a_n e(nx) with short intervals in the index n. The paper writes the short interval as ∑_{x<n≤x+h} χ^k(n)Λ(n)e^{-n^k/N}, which is not what the standard lemma yields. A change of variable to m=n^k would give intervals in m, not in n. No statement or proof of the needed adaptation is provided. This is load-bearing for the entire I2 estimate in (3.2)–(3.13). Please supply a proof of the claimed variant, or replace this step with a valid large-sieve bound for polynomial phases.
- [§3.5, Eq. (3.13)] Even granting the Gallagher lemma adaptation, the estimate (3.13) gives I2(N,h)≪hN^{1/k}log^2N + h^2log^2(2q). After dividing by h^2 and substituting into the dyadic sum in (3.2), the h^2log^2(2q) term contributes N log^2 q (since the outer weight is N/2^m and summing over m gives N log^2 q). This is not dominated by N^{1/k}log^3N for k≥2. The sentence 'q-terms do not appear because q is fixed and thus its contribution can be neglected' is not valid: a constant factor is not negligible when it is not multiplied by a term that vanishes. Thus the claimed bound I2=O(N^{1/k}log^3N) is not established.
- [§3.2, Eqs. (2.7) and (3.1)] The main term I1 is defined in (2.7) with |Ψ_k(z,χ0)|^2, but Lemma 2.1 gives a product Ψ_k(z,χ^k)Ψ_k(z,\bar{χ}^k), which for χ^k=χ0 equals Ψ_k(z,χ0)^2, not its absolute square. In (3.1), the paper replaces this by (Ψ_k(z)+O(...))^2, which is only consistent with the non-absolute square. If the absolute value is retained, the integral does not equal G_{1,k}(N) because the phase in the generating function is additive (n^k+m^k), not the difference. Please correct the notation and the main-term derivation.
- [Theorem 1.3 vs. §3.5] The proof concludes at the end of §3.5 that I2=O(N^{1/k}log^3N). This does not match the error term in Theorem 1.3, which is O(N^{1/k}log^2N log q/φ(q)). For fixed q, log^3N is not O(log^2N). Even if the rest of the proof were correct, the final error would be at least O(N^{1/k}log^3N). Either the theorem's error term must be weakened to O(N^{1/k}log^3N), or the I2 estimate must be improved by a factor log N.
- [Lemma 3.1] The proof of Lemma 3.1 for the case where χ^k is not primitive is deferred to 'arguing like in the proof of Lemma 2.4 of [4]' without providing the details. This lemma is essential for the estimates of J1 and J2, which directly feed into the short-interval bounds. Since the case χ^k=χ0 is excluded, but χ^k may still be non-primitive, a complete proof is needed. Please supply the full argument or a precise reference with the necessary modifications.
minor comments (4)
- [References] Reference [1]: 'Springler-Verlag' should be 'Springer-Verlag'.
- [Lemma 4.1, p=2 case] In the statement for p=2, α≥3, the condition '3≤β<α−2' appears to be a typo. For example, β=1 or 2 with α≥4 should also be covered; the proof text uses '1≤β<α−2'. Please correct the range.
- [§3.5, Eq. (3.11)] In the integration by parts after (3.11), the function J1(x) is defined using ψ_k(x,χ), while the integral contains ψ_k(x+h,χ). The integration by parts should involve the shifted version J1^{(h)}(x)=∫_0^x |ψ_k(u+h,χ)|^2 du, not J1(x). This is not justified and needs correction.
- [Sketch of proof] The displayed bound 'I2 ≪ N^{1/k} log^3 N log^2 q < N^{1/k+1}' is confusing; the first inequality is far weaker than the final claim in Theorem 1.3. Please align the sketch with the actual proof.
Circularity Check
No circularity: the q-average is expanded by orthogonality into the unrestricted average times an independently evaluated character sum; the flagged Gallagher gap is a correctness issue, not circularity.
full rationale
The derivation is self-contained and does not reduce to its own inputs. The main term in Theorem 1.3 is obtained by writing the congruence condition q | m_1^k + m_2^k via character orthogonality (Lemma 2.1, equation (2.4)). The contribution of the principal characters, i.e. those with chi^k = chi_0, gives I_1 = (Sigma_k(q)/phi(q)) G_{1,k}(N) plus an error, because by Lemma 2.2 the principal character sum Psi_k(z, chi_0) equals the unrestricted Psi_k(z) up to a small error. No parameter is fitted to G_{q,k}(N); Sigma_k(q) is evaluated independently in Lemma 4.1 from the cyclic structure of the multiplicative group modulo q and is not obtained from the target average. The cited results [2], [3], [4], and [5] are prior lemmas with proofs or proof sketches given here, and the one self-citation to [5] (Languasco-Zaccagnini) is not load-bearing in the sense of being unverified or assumed in place of proof. The only substantive concern in the paper, the adaptation of Gallagher's lemma in Section 3.3 from frequencies n to n^k, is a technical correctness gap, not a circularity; even if that estimate failed, the main-term identity would remain a structural expansion, not a definitional tautology. Therefore no circular step is present.
Assumptions & free parameters
assumptions (4)
- domain assumption GRH for Dirichlet L-functions modulo q
- ad hoc to paper Gallagher's lemma (Montgomery Lemma 1.9) applies with short intervals over n for frequencies n^k
- ad hoc to paper Lemma 3.1 holds for nonprimitive chi^k by an argument 'like in the proof of Lemma 2.4 of [4]'
- standard math Standard Perron/explicit formula and residues for L'/L
Cite this review
Pith. "Pith review of The average number of representations of an integer as a sum of two prime powers over multiples of a fixed integer." pith.science (2026). https://pith.science/paper/WJFZZHQG
@misc{pith2026260324120,
author = {Pith},
title = {Pith review of: The average number of representations of an integer as a sum of two prime powers over multiples of a fixed integer},
year = {2026},
howpublished = {\url{https://pith.science/paper/WJFZZHQG}},
note = {Machine review of arXiv:2603.24120}
}
abstract
We extend a result by Ikeda and Suriajaya (2025) to find the asymptotic behaviour of the average number of representations of an integer $n$, over multiples of a fixed $q\ge 2$, as a sum of two prime $k$-th powers, for $k\ge 2$.
Reference graph
Works this paper leans on
-
[7]
H. L. Montgomery.Topics on multiplicative number theory. Lecture Notes in Mathematics. Springer-Verlag, Berlin-Heidelberg-New York, 1971. 17
1971
-
[4]
Ikeda and A
I. Ikeda and A. I. Suriajaya. The average number of Goldbach representa- tions over multiples ofq.Funct. Approx. Comment. Math. Advance Publication, 2(73):169–183, 2025
2025
-
[1]
Davenport.Multiplicative number theory
H. Davenport.Multiplicative number theory. Number 74 in Graduate Texts in Mathematics. Springler-Verlag, 3rd edition, New York, 2000
2000
-
[2]
Goldston and A
D.A. Goldston and A. I. Suriajaya. On a smoothed version of the number of Goldbach representations.Number Theory in Memory of Eduard Wirsing, H. Maier et al. (eds.), Springer Nature Switzerland AG, pages 145–146, 2023
2023
-
[3]
Goldston and R.C
D.A. Goldston and R.C. Vaughan. On the Montgomery-Hooley asymptotic formula, Sieve Methods, Exponential Sums, and their Application in Number Theory. Cambridge University Press, greaves, harman, huxley eds. edition, 1996
1996
-
[5]
Languasco and A
A. Languasco and A. Zaccagnini. Sums of many primes.J. Number Theory, 132:1265–1283, 2012
2012
-
[6]
Migliaccio and A
A. Migliaccio and A. Zaccagnini. The average number of representation of an integer as a sum of at least three primes over multiples of a fixed integer. Work in progress
Reviewed August 2, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.