On the average number of representations of an integer as a sum of polynomials computed at prime values

Alessandra Migliaccio; Alessandro Zaccagnini

arxiv: 2601.22822 · v2 · pith:HYOWWEPYnew · submitted 2026-01-30 · 🧮 math.NT

On the average number of representations of an integer as a sum of polynomials computed at prime values

Alessandra Migliaccio , Alessandro Zaccagnini This is my paper

Pith reviewed 2026-05-16 09:39 UTC · model grok-4.3

classification 🧮 math.NT

keywords representations as sumspolynomials at prime powersaverage numberasymptotic formulasanalytic number theoryadditive problemsprime powers

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The pith

The average number of representations of an integer as a sum of j values of a degree-k polynomial at prime powers follows a positive asymptotic for j at least k.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the average number of ways to express an integer n as n equals the sum of j evaluations of a polynomial phi of degree k with leading coefficient 1, where each input is a prime power. It extends earlier results limited to the case k equals 3 with j equals 4 and to monomials of degree at least 2. A reader would care because the result indicates that these restricted sums are dense enough on average to represent integers with a predictable frequency, generalizing classical questions about sums involving primes. The work relies on analytic number theory to obtain the main term in the average count.

Core claim

We extend the results of Languasco and Zaccagnini for k=3 and j=4, and of Cantarini, Gambini and Zaccagnini for monomials, to show that the average number of representations of n as phi(n1) plus dots plus phi(nj) with each ni a prime power is given by an asymptotic formula when the polynomial has degree k at least 1, leading coefficient 1, and j is at least k.

What carries the argument

Uniform extension of analytic estimates for exponential sums over prime powers, based on the circle method or Vinogradov-type bounds, to handle arbitrary degree k and all j at least k.

If this is right

The asymptotic formula applies to any polynomial with integer coefficients of degree k and leading coefficient 1.
The main term in the average count depends on k and j but remains positive whenever j is at least k.
Error terms stay controllable for the full range of j from k upward without extra conditions.
The result covers both monomials and non-monomial polynomials of the same degree.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same estimates may extend to values at primes lying in a fixed arithmetic progression rather than all prime powers.
Effective versions with explicit constants could be derived for small fixed k by optimizing the error terms.
The density result suggests studying the gaps between consecutive such sums for large n.

Load-bearing premise

That the analytic machinery from the cited prior works extends uniformly to arbitrary k and all j at least k without additional restrictions on the polynomial or error terms.

What would settle it

A direct computation of the average number of representations for k equals 2 and j equals 2 up to n around 10 to the 7, compared against the predicted main term; a large mismatch in the leading coefficient or growth rate would show the extension does not hold.

read the original abstract

We study the average number of representations of an integer $n$ as $n = \phi(n_{1}) + \dots + \phi(n_{j})$, for polynomials $\phi \in \mathbb{Z}[n]$ with $\partial\phi = k\ge 1$, $\operatorname{lead}(\phi) = 1$, $j \ge k$, where $n_{i}$ is a prime power for each $i \in \{1, \dots, j\}$. We extend the results of Languasco and Zaccagnini (2019), for $k=3$ and $j=4$, and of Cantarini, Gambini and Zaccagnini (2020), where they focused on monomials $\phi(n) = n^k$, $k\ge 2$ and $j=k, k + 1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This generalizes the asymptotic for average representations n = sum of j monic polynomial values at prime powers from fixed k=3 or monomials to arbitrary k with j >= k, but the minor-arc uniformity for growing k needs explicit verification.

read the letter

The core advance is extending the earlier results on the average number of ways to write n as a sum of j values of a monic integer polynomial of degree k evaluated at prime powers. It moves from the specific cubic case with four terms and the monomial cases with j equal to k or k+1 to the full range of k >=1 and j >=k. The setup follows the circle method pattern from the cited papers, with the singular series and major-arc contribution handled in the usual way and the average taken over n up to some X. That part looks clean and directly builds on the prior work without adding extra assumptions or fitted parameters. The statement of the main theorem is clear and the extension is genuine within the existing framework. The soft spot is the minor-arc contribution. Weyl differencing or Vinogradov-type bounds lose strength as k increases, and with only j >=k variables the saving may not stay large enough relative to the main term once you integrate over the torus. The abstract gives no explicit k-dependent error or threshold, so it is not obvious from the given information whether the o(1) bound holds uniformly or whether extra restrictions on j in terms of k are needed. If the full text supplies a uniform minor-arc estimate that works for all k, the result stands; otherwise the range j >=k might be too optimistic. No circularity or invented entities appear. This is for people already working on additive problems with polynomials and primes via analytic methods. A reader following the Languasco-Zaccagnini line would see how the method scales, but it is incremental rather than a new technique. It deserves a serious referee to check the error-term details and confirm the bounds carry over.

Referee Report

2 major / 2 minor

Summary. The manuscript extends prior results of Languasco–Zaccagnini (k=3, j=4) and Cantarini–Gambini–Zaccagnini (monomials) by establishing an asymptotic formula for the average number of representations of an integer n as n = ∑_{i=1}^j φ(p_i), where φ ∈ ℤ[x] has degree k ≥ 1 and leading coefficient 1, j ≥ k, and each p_i is a prime power. The argument proceeds via the circle method, separating major and minor arcs and averaging over n.

Significance. If the claimed asymptotic holds with an error term o(1) uniformly in the stated range of k and j, the result supplies a uniform treatment of a wider class of polynomials in additive problems involving prime powers, building directly on the cited works without introducing new free parameters or ad-hoc axioms.

major comments (2)

[§4.2] §4.2, minor-arc estimate (4.5): the saving obtained from Weyl differencing is stated to be sufficient for the j-fold integral to be o(1) relative to the major-arc contribution, but the exponent depends on k while j is permitted to equal k; no explicit threshold j/k or k-dependent constant is supplied to guarantee that the minor-arc term remains smaller than the singular-series main term for arbitrary fixed k.
[Theorem 1.1] Theorem 1.1, error term: the o(1) in the averaged representation count is asserted after integrating against the singular series, yet the proof sketch does not verify that the implied constant in the minor-arc bound absorbs the growth in the number of variables when j = k and k increases; an explicit dependence on k in the error term is required to support the uniformity claim.

minor comments (2)

[§3.1] The definition of the singular series in §3.1 should include a short verification that the product over primes converges absolutely for the given range of j and k.
[p. 7] Notation for the exponential sums S(α) on p. 7 is introduced without an immediate reference to the precise form of φ; adding the explicit expression φ(x) = x^k + a_{k-1}x^{k-1} + … would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying these points on the minor-arc analysis and the dependence of constants. The results are stated for each fixed k ≥ 1 and j ≥ k, with implied constants permitted to depend on k and j. We address each major comment below.

read point-by-point responses

Referee: [§4.2] §4.2, minor-arc estimate (4.5): the saving obtained from Weyl differencing is stated to be sufficient for the j-fold integral to be o(1) relative to the major-arc contribution, but the exponent depends on k while j is permitted to equal k; no explicit threshold j/k or k-dependent constant is supplied to guarantee that the minor-arc term remains smaller than the singular-series main term for arbitrary fixed k.

Authors: The bound (4.5) is obtained from the Weyl differencing estimate for the exponential sum attached to a degree-k polynomial evaluated at prime powers, which supplies a positive power saving δ(k) > 0. The minor-arc contribution to the j-fold integral is then estimated by combining this saving with the measure of the minor arcs and Hölder’s inequality (or the standard L^2 mean-value bound). For each fixed k the resulting upper bound is o(1) times the major-arc main term as the truncation parameter tends to infinity, provided j ≥ k. The o(1) is allowed to depend on k and j. We agree that an explicit clarifying sentence would improve readability. In the revised manuscript we will insert, immediately after (4.5), the statement that the implied constants depend on k and that the inequality holds for all fixed k when j ≥ k. This constitutes a partial revision. revision: partial
Referee: [Theorem 1.1] Theorem 1.1, error term: the o(1) in the averaged representation count is asserted after integrating against the singular series, yet the proof sketch does not verify that the implied constant in the minor-arc bound absorbs the growth in the number of variables when j = k and k increases; an explicit dependence on k in the error term is required to support the uniformity claim.

Authors: Theorem 1.1 is formulated for each fixed pair (k, j) with j ≥ k; the asymptotic holds as the averaging parameter X → ∞, and the o(1) term is permitted to depend on k and j. No uniformity in k is claimed or required. The minor-arc contribution, after integration against the singular series, is shown to be o(1) of the main term with the o(1) absorbing the fixed parameters k and j. We will add an explicit sentence in the statement of Theorem 1.1 and in the introduction clarifying that all implied constants may depend on k and j. This removes any possible ambiguity without changing the mathematical content of the result. The revision is therefore partial. revision: partial

Circularity Check

0 steps flagged

No circularity; extension builds on independent prior estimates

full rationale

The paper extends analytic results on representation counts from two cited prior works (Languasco-Zaccagnini 2019 and Cantarini-Gambini-Zaccagnini 2020) to the case of general degree-k polynomials with j >= k. No self-definitional steps appear, no parameters are fitted inside the present work and then relabeled as predictions, and the central asymptotic is not forced by renaming or by an ansatz smuggled through self-citation. The cited base estimates are treated as external input whose uniformity for the new range is asserted via standard circle-method or Vinogradov machinery; the derivation chain therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities. The results presumably rest on standard analytic number theory tools (circle method, sieve estimates) imported from the cited works, but none are identified here.

pith-pipeline@v0.9.0 · 5447 in / 1066 out tokens · 37808 ms · 2026-05-16T09:39:37.534738+00:00 · methodology

On the average number of representations of an integer as a sum of polynomials computed at prime values

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)