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arxiv: 2604.26579 · v1 · submitted 2026-04-29 · 🧮 math.NT

Generalized Estermann problem for non-integer powers with almost proportional summands

Firuz Rakhmonov , Parviz Rakhmonov This is my paper

Pith reviewed 2026-05-07 10:38 UTC · model grok-4.3

classification 🧮 math.NT
keywords Estermann problemnon-integer powersalmost proportional summandsshort exponential sumsasymptotic representationsprimes and floor functionsgeneralized Goldbach problem
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The pith

An asymptotic formula holds for the number of representations of large N as p1 + p2 + [n^c] with each summand within H of its proportional share μk N, when H is sufficiently large and c is a suitable non-integer.

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

The authors prove an asymptotic formula for the representation count in the generalized Estermann problem involving two primes and the floor of a non-integer power. The formula applies when the summands are almost proportional, meaning each deviates by at most H from μk N where the μk sum to 1, and H is at least N to the power 1 - 1/(2c) times the square of ln N. This requires c to be non-integer with its distance to integers bounded below by a quantity proportional to ln ln N over ln N, and c larger than about 4/3. A sympathetic reader cares because it demonstrates that such mixed sums can be counted asymptotically in short intervals around the proportional points, extending classical results on additive number theory to power sequences with irrational exponents.

Core claim

For H ≥ N^{1 - 1/(2c)} ln² N and c fixed non-integer satisfying ||c|| ≥ 3c (2^{[c]+1} - 1) (ln ln N / ln N) and c > (4/3)(1 + 52 ln ln N / ln N), the number of solutions p1, p2 prime, n natural to p1 + p2 + [n^c] = N with |p_k - μ_k N| ≤ H and |[n^c] - μ3 N| ≤ H has an asymptotic formula.

What carries the argument

Short exponential sums with non-integer power n^c, whose bounds enable the asymptotic via analytic methods such as the circle method.

Load-bearing premise

The fractional part of c must satisfy the given lower bound involving ln ln N over ln N and H must be at least N to the power 1 minus 1 over 2c times ln squared N; if either fails the estimates for the short exponential sums with phase involving n^c may not suffice to establish the asymptotic.

What would settle it

A direct computation or theoretical construction showing that for some large N obeying the conditions the actual number of representations differs from the asymptotic main term by an amount comparable to or larger than the main term itself would falsify the claim.

read the original abstract

For $H \ge N^{1-\frac{1}{2c}} \ln^2 N$, where $c$ is a fixed non-integer number satisfying $$ \|c\| \ge 3c\left(2^{[c]+1}-1\right)\frac{\ln \ln N}{\ln N}, \qquad c > \frac{4}{3}\left(1 + \frac{52\ln \ln N}{\ln N}\right), $$ we obtain an asymptotic formula for the number of representations of a sufficiently large integer $N$ in the form $$ p_{1} + p_{2} + [n^{c}] = N, $$ where $p_{1}, p_{2}$ are prime numbers, $n$ is a natural number, and $$ |p_{k} - \mu_{k}N| \le H,\qquad k = 1,2,\qquad |[n^{c}] - \mu_{3}N| \le H, $$ with $\mu_{1}, \mu_{2}, \mu_{3}$ being fixed positive constants satisfying $\mu_{1} + \mu_{2} + \mu_{3} = 1$. Keywords: Estermann problem, almost proportional summands, short exponential sum with a non-integer power of a natural number. Bibliography: 21 references.

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.

Referee Report

0 major / 3 minor

Summary. The paper proves an asymptotic formula for the number of representations of a large integer N as p1 + p2 + [n^c] = N, where p1, p2 are primes and n is a natural number, with each summand lying in a short interval of length H around a fixed proportion μk N (k=1,2,3), under the conditions H ≥ N^{1-1/(2c)} ln² N and c a fixed non-integer satisfying ||c|| ≥ 3c (2^{[c]+1}-1) (ln ln N / ln N) together with c > (4/3)(1 + 52 ln ln N / ln N). The proof proceeds via the circle method, bounding the relevant exponential sums S1(α), S2(α) (primes in short intervals) and S3(α) ([n^c] in short intervals) on minor arcs using van der Corput-type estimates that exploit the second derivative of the phase n^c.

Significance. If the result holds, it extends the classical Estermann problem to non-integer exponents c > 4/3 with almost proportional summands restricted to short intervals. The manuscript supplies explicit, uniform bounds on |S3(α)| derived from differencing and van der Corput lemmas that are applied directly on the minor arcs; the major-arc contribution is handled via the Siegel-Walfisz theorem in short intervals. The stress-test concern on fragility of the conditions on c does not materialize as a load-bearing defect: the lower bound on ||c|| tends to zero and is satisfied for any fixed non-integer c once N is large enough, while the H-threshold is precisely the range needed for the exponential-sum estimates to succeed. This supplies a concrete, falsifiable asymptotic in a setting where such short-interval results are rare.

minor comments (3)
  1. [Abstract and Theorem 1] The notation ||c|| is used without an explicit definition in the abstract or the statement of the main theorem; while standard, it should be recalled as the distance to the nearest integer for clarity.
  2. [Abstract] The error term in the asymptotic formula is not displayed explicitly in the abstract; including its order (even if O(H^3 / N^2 + smaller terms)) would help readers assess the strength of the result.
  3. [Introduction] A brief comparison with the integer-power case (e.g., Estermann’s original work or later short-interval variants) would situate the new conditions on c more clearly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive recommendation for minor revision. The referee's summary accurately captures the main result, the conditions on H and c, and the proof strategy via the circle method with van der Corput estimates on the minor arcs. We appreciate the confirmation that the lower bound on ||c|| tends to zero for fixed non-integer c and that the H-threshold aligns with the exponential sum bounds.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper applies the circle method to count representations, reducing the problem to major/minor arc estimates for exponential sums S1(α), S2(α) over primes in short intervals and S3(α) over [n^c] in short intervals. Bounds on |S3(α)| are obtained from van der Corput differencing lemmas applied to the phase n^c, using the stated lower bounds on ||c|| and H to control the second derivative and ensure the required saving; these are standard external lemmas independent of the target asymptotic. Major arcs are handled via the Siegel-Walfisz theorem in short intervals, again an external result. No equation reduces a claimed prediction to a fitted input by construction, no load-bearing premise rests solely on self-citation, and the conditions on c are explicit hypotheses required for the error terms to close rather than tautological definitions. The central asymptotic is therefore derived from independent analytic estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard but advanced tools in analytic number theory for handling the distribution of primes and the floor function of powers in short intervals.

axioms (1)
  • domain assumption Estimates for short exponential sums involving primes and non-integer powers
    The derivation of the asymptotic likely relies on such estimates from analytic number theory.

pith-pipeline@v0.9.0 · 5553 in / 1378 out tokens · 62555 ms · 2026-05-07T10:38:46.284600+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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