An Explicit Surjectivity Threshold for Digit Sums of Primes

Jens Lehmann

REVIEW 2 major objections 2 minor 16 references

An Explicit Surjectivity Threshold for Digit Sums of Primes

T0 review · 2 major / 2 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Every integer m at least M < 1.78 × 10^32 and coprime to 9 is the digit sum of some prime.

desk verdict The paper turns the DMR circle-method framework into the first explicit numerical threshold M < 1.78e32 for digit-sum surjectivity on primes, with a clean application to an OEIS sequence. read the letter →

arxiv 2606.04677 v1 pith:TP5KQJWO submitted 2026-06-03 math.NT

Jens Lehmann This is my paper · ORCID

classification math.NT MSC 11N0511A63

keywords digitsumprimessurjectivitycirclemethodexplicitconstantsnumbertheoryadditiveproblems

topics The Twin Prime Conjecture

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 makes explicit the constants in the Drmota-Mauduit-Rivat circle method to prove that digit sums of primes are surjective onto all sufficiently large integers coprime to 9. It produces a concrete bound M below 1.78 times 10 to the 32 after which every such m appears as the sum of decimal digits of at least one prime. This turns a known existence result into an effective statement with a numerical threshold. The work also gives an explicit lower bound on the number of primes with given digit sum and applies it to show there are infinitely many primes whose digit sum chain stays prime.

What carries the argument

The circle method framework with explicit versions of major-arc estimates and Type-II minor-arc estimates, replacing DMR's implicit prime exponential sums.

What would settle it

An integer m larger than or equal to M with gcd(m,9)=1 for which no prime p has digit sum equal to m would falsify the claim, or a verification that the error terms in the asymptotic formula exceed the main term for some range.

Watch

Extended reading notes

Core claim

We exhibit an explicit integer M < 1.78 × 10^{32} such that every integer m ≥ M with gcd(m,9)=1 occurs as s(p) for at least one prime p. The proof uses explicit major-arc estimates, a fully explicit replacement for the implicit prime exponential-sum input, and constant-tracked Type-II minor-arc estimates.

Load-bearing premise

The explicit major-arc estimates, the replacement for the prime exponential-sum input, and the Type-II minor-arc estimates are accurate enough to give the stated error terms and the numerical value of M.

Editorial extensions

If this is right

Infinitude of the sequence of primes whose iterated decimal digit sums remain prime down to a single digit prime.
Explicit lower bound A_m(10^{2m/9}) ≥ C_q(m) 10^{2m/9} / m^{3/2} with C_q(m) positive above the threshold.
Effective statements for additive primes and digit sum decompositions.
Every admissible residue class modulo 9 is eventually hit by digit sums of primes above the threshold.

Reading between the lines

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

This approach of tracking constants explicitly could be applied to other problems in additive number theory involving primes and digital conditions.
Improving the bound on M would require sharper explicit estimates in the minor arcs or better exponential sum bounds.
The existence of such an M implies that the set of digit sums of primes has positive density in the admissible classes for large values.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The key point is that this work supplies the first published explicit numerical bound: every sufficiently large m coprime to 9 appears as the decimal digit sum of at least one prime, with the threshold pinned at M below 1.78 times 10 to the 32. They achieve this by replacing the implicit inputs in the Drmota-Mauduit-Rivat setup with fully explicit versions, including the prime exponential-sum bound, major-arc estimates, and constant-tracked Type-II minor-arc estimates.

What the paper does well is carry the bookkeeping all the way through to a concrete number and then apply it to prove infinitude for the primes in OEIS A070027, where the iterated digit-sum chain remains prime until it hits a single-digit prime. The explicit lower bound on the counting function A_m is also stated with a positive constant C_q(m) that stays away from zero in admissible classes. This is the sort of effective consequence that the original asymptotic theory points toward but rarely delivers with numbers.

The soft spot is the size of M itself. Even if every constant is tracked correctly, a threshold at 10^32 has almost no computational reach, so the result stays largely theoretical. The load-bearing step is the accuracy of those explicit replacements and error-term calculations; a modest underestimate in one decay rate could shift the final M by orders of magnitude, though the underlying asymptotic framework would remain intact. The paper does not appear to introduce circularity or fitted parameters.

This is specialized work aimed at people who care about effective versions of analytic number theory results on digital restrictions of primes. A serious referee should see it because the explicitness is the actual contribution, and the calculations are checkable in principle even if the bound is large. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper extends the circle-method framework of Drmota-Mauduit-Rivat on digital restrictions for primes by making all relevant constants explicit. It proves there exists an explicit integer M < 1.78 × 10^{32} such that every integer m ≥ M with gcd(m,9)=1 is realized as the decimal digit sum s(p) of at least one prime p. It also establishes an explicit lower bound A_m(10^{2m/9}) ≥ C_q(m) 10^{2m/9}/m^{3/2} with C_q(m) positive and bounded away from zero in admissible classes, and derives consequences including the infinitude of OEIS A070027.

Significance. If the explicit constant derivations hold, the result supplies the first published numerical surjectivity threshold for digit sums of primes (previously known only to exist by Harman via the DMR asymptotics). The explicit lower bound and applications to additive primes and iterated digit-sum chains are direct consequences of the same estimates.

major comments (2)

[Type-II minor-arc estimates] Abstract and § on Type-II minor-arc estimates: the numerical value M < 1.78 × 10^{32} is load-bearing on the fully explicit replacement for DMR's prime exponential-sum bound and the constant-tracked Type-II estimates; the manuscript must supply the complete error-term derivations and numerical verifications of those constants (including all implicit factors from the circle-method major/minor arc decompositions) so that the bound on M can be independently checked.
[major-arc estimates and prime exponential-sum replacement] The explicit major-arc estimates and the replacement for the DMR prime exponential-sum input are asserted to suffice for the stated error terms, but without tabulated intermediate constants or a verification appendix the specific numerical threshold cannot be confirmed; any underestimation of an implicit constant would invalidate the concrete M even if the asymptotic framework is correct.

minor comments (2)

[lower bound statement] Notation for C_q(m) should be defined explicitly in the statement of the lower bound rather than only in the surrounding text.
[applications] The application to OEIS A070027 would benefit from a short explicit statement of the iterated digit-sum condition in terms of the surjectivity result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for highlighting the need for full verifiability of the explicit constants. We agree that the numerical threshold M requires complete, checkable derivations of all error terms. Below we respond point-by-point and commit to adding a dedicated verification appendix in the revised manuscript.

read point-by-point responses

Referee: [Type-II minor-arc estimates] Abstract and § on Type-II minor-arc estimates: the numerical value M < 1.78 × 10^{32} is load-bearing on the fully explicit replacement for DMR's prime exponential-sum bound and the constant-tracked Type-II estimates; the manuscript must supply the complete error-term derivations and numerical verifications of those constants (including all implicit factors from the circle-method major/minor arc decompositions) so that the bound on M can be independently checked.

Authors: We agree that independent verification of M requires the full set of explicit error-term derivations. The manuscript already replaces DMR's implicit prime exponential-sum bound with an explicit version and tracks constants through the Type-II estimates, but the intermediate numerical factors from the circle-method decompositions are not collected in one place. We will add a new appendix that derives every error term from first principles, lists all implicit constants with their numerical values, and recomputes the final bound on M step-by-step. revision: yes
Referee: [major-arc estimates and prime exponential-sum replacement] The explicit major-arc estimates and the replacement for the DMR prime exponential-sum input are asserted to suffice for the stated error terms, but without tabulated intermediate constants or a verification appendix the specific numerical threshold cannot be confirmed; any underestimation of an implicit constant would invalidate the concrete M even if the asymptotic framework is correct.

Authors: The major-arc estimates and the explicit replacement for the DMR prime exponential sum are stated with explicit forms in the text. We acknowledge, however, that without a tabulated list of all intermediate constants the numerical threshold cannot be independently confirmed. The revision will therefore include, in the same new verification appendix, a table of every constant appearing in the major-arc analysis together with its derivation and numerical value. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: explicit estimates extend external DMR framework

full rationale

The paper builds on the prior DMR circle-method results (distinct authors) to derive explicit constants for major-arc estimates, a fully explicit prime exponential-sum bound, and constant-tracked Type-II minor-arc estimates, yielding the numerical surjectivity threshold M. No step reduces by the paper's own equations to a fitted input, self-definition, or self-citation chain; the central claim is the output of these independent explicit derivations rather than a renaming or tautological prediction. The derivation is self-contained against the external DMR asymptotics.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the accuracy of explicit versions of the Drmota-Mauduit-Rivat circle-method estimates; no free parameters or new entities are introduced.

assumptions (1)

domain assumption The Drmota-Mauduit-Rivat circle-method framework for digital restrictions on primes admits fully explicit major-arc and minor-arc estimates with the stated error terms.
The paper invokes this framework to obtain the explicit surjectivity threshold.

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Pith. "Pith review of An Explicit Surjectivity Threshold for Digit Sums of Primes." pith.science (2026). https://pith.science/paper/TP5KQJWO

@misc{pith2026260604677,
  author       = {Pith},
  title        = {Pith review of: An Explicit Surjectivity Threshold for Digit Sums of Primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TP5KQJWO}},
  note         = {Machine review of arXiv:2606.04677}
}

abstract

Let $s(n)=s_{10}(n)$ be the decimal sum-of-digits map. Building on the circle-method framework of Drmota-Mauduit-Rivat for digital restrictions on primes, we make the constants explicit at the points needed to obtain an effective surjectivity statement for digit sums of primes. We exhibit an explicit integer $M < 1.78 \times 10^{32}$ such that every integer $m \ge M$ with gcd$(m,9)=1$ occurs as $s(p)$ for at least one prime $p$. We also prove an explicit lower bound $A_m(10^{2m/9}) \ge C_q(m) 10^{2m/9}/m^{3/2}$, where $C_q(m)$ is explicit, positive above the sufficient threshold, and bounded away from $0$ along each admissible residue class. Existence of a non-numerical threshold follows from the DMR asymptotic theory and was noted by Harman; to the best of our knowledge, this is the first published explicit numerical threshold for this surjectivity statement. The proof combines explicit major-arc estimates, a fully explicit replacement for DMR's implicit prime exponential-sum input, and constant-tracked Type-II minor-arc estimates. As an application, we prove the infinitude of OEIS A070027, the primes whose iterated digit-sum chain remains prime until reaching a one-digit prime. We also record related effective consequences for additive primes and digit-sum additive decompositions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.

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