The Prime Digit Distribution Conjecture: A Formal Proof of Average Digit Equidistribution in the Prime Numbers

Mahadee Al Mobin; Md. Shariful Islam

REVIEW 2 major objections 4 minor 21 references

The Prime Digit Distribution Conjecture: A Formal Proof of Average Digit Equidistribution in the Prime Numbers

T0 review · 2 major / 4 minor · reviewed 2026-07-14 · grok-4.5

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

Pith's one-line read Pooled digits of primes below 10^n are equidistributed: each digit occurs with frequency 1/10 plus an O((log n)/n) error.

desk verdict Averaged digit equidistribution for primes is a clean, modest new statement; the architecture works, but Lemma 3.2 needs a proper short-interval unweighted bound before the proof is complete. read the letter →

arxiv 2607.10654 v1 pith:5WZDB6NC submitted 2026-07-12 math.NT

Mahadee Al Mobin , Md. Shariful Islam This is my paper · ORCID

classification math.NT MSC 11A6311N0511N1311K3811L07

keywords sumofdigitsprimedigitdistributionequidistributionexponentialsumsErdős–TuráninequalityVaughanestimatesdiscrepancytheorydecimalprimes

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 claims that if you collect every decimal digit from every prime smaller than 10^n and count how often each digit 0 through 9 appears, the frequency of each digit is 1/10 plus an error that shrinks like (log n)/n. The argument works by first proving that almost every digit position inside an m-digit prime shell is equidistributed, then showing that the O(log m) positions near the leading and trailing ends contribute a vanishing share of the total digit mass, and finally summing the shells using the geometric growth of prime counts. A sympathetic reader cares because this is an unconditional, averaged form of the long-standing intuition that primes look random in their digits, obtained from classical tools without any unproved hypotheses. The authors carefully separate this pooled result from the still-open questions of normality in a fixed position and of joint digit statistics.

What carries the argument

The Interior Digit Lemma: for positions k that lie between C1 log m and m - C2 log m inside an m-digit prime shell, the count of primes with k-th digit equal to d equals π_m/10 plus an error O(π_m/(log Y)^A). It is proved by feeding classical Vaughan–Vinogradov bounds on exponential sums over primes into the Erdős–Turán discrepancy inequality, then diluting the O(log m) exceptional end positions by averaging.

What would settle it

Compute the empirical frequencies of digits 0–9 among all primes below 10^n for successive large n (say up to 10^12 or higher) and check whether the maximal deviation from 1/10 decays at least as fast as (log n)/n; a persistent larger deviation would refute the claimed error term.

Watch

Extended reading notes

Core claim

For every digit d from 0 to 9, the total number of occurrences of d among all digits of all primes less than 10^n, divided by the total number of such digits, equals 1/10 plus an error of size O((log n)/n) as n tends to infinity, uniformly in d. The same limit therefore holds in total variation for the empirical digit measure.

Load-bearing premise

The argument stands or falls on a classical bound for exponential sums over the primes in each m-digit shell; if that bound fails for the moduli that encode interior digit positions, the interior equidistribution step collapses.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The one thing to know: this is a genuine averaged equidistribution theorem for individual decimal digits of primes (pooled over all places and all p < 10^n) with an explicit O((log n)/n) rate. That statement is not in Mauduit–Rivat, Maynard, or the other digit-sum literature they cite, and the authors correctly flag that it is weaker than pointwise normality or joint digit laws.

What they do well is the global plan. Interior places are handled by Erdős–Turán plus classical exponential sums; an O(log m) edge band is diluted by trivial bounds; geometric growth of shell mass lets the top shell dominate. The conclusion section is unusually honest about scope. Circularity is zero—everything rests on PNT, Erdős–Turán, and Vinogradov-type estimates from Davenport. The rate is elementary but clean.

The soft spot is real and load-bearing, though fixable. Lemma 3.2 claims the unweighted shell sum S_m(h,q) inherits the classical majorant (Y/√r + Y^{4/5} + √(Y r))(log Y)^4. The write-up compares it to the weighted cumulative sum with a literal “<” and never writes the short-interval difference or partial summation. That gap sits under the interior-digit lemma and therefore under the main theorem. Standard tools (partial summation + Vaughan identity on [Y/10,Y]) almost certainly close it, but as printed the estimate is incomplete. Two self-citations ([18,19]) are also mis-described as surveys of the digit problem; that is sloppy but peripheral.

This is for analytic number theorists who care about digital statistics of primes. It is not a breakthrough, but it is a coherent unconditional corollary that the literature does not already contain. A serious editor should send it to referees; the repairs are routine and the claim is worth having on the record once the exponential-sum step is written properly. I would not cite it until that is fixed, but I would bring the cleaned version to reading group.

Referee Report

2 major / 4 minor

Summary. The paper claims an unconditional averaged equidistribution theorem for the decimal digits of primes: if one pools every digit position of every prime less than 10^n, the frequency of each digit d equals 1/10 + O((log n)/n) uniformly in d (Theorem 2.2). The argument proceeds by shells of m-digit primes, applies the Erdős–Turán inequality to the fractional parts {p/10^{k+1}} at each position k, bounds the resulting exponential sums over the shell via a classical Vinogradov-type majorant, obtains a power-saving error for all but O(log m) exceptional positions near the ends, dilutes those positions by a trivial estimate, and finally sums across shells using the geometric growth of shell mass under the Prime Number Theorem.

Significance. If the estimates are made rigorous, the result supplies a clean, elementary proof of the natural averaged form of digit equidistribution for primes, with an explicit power-saving error and a careful distinction from the still-open pointwise, normality, and correlation questions. The architecture (interior discrepancy + edge dilution + geometric domination) is transparent and uses only classical tools, so the paper would be a useful reference clarifying the precise scope of what is currently known. It does not supersede the deeper Mauduit–Rivat theory for sum-of-digits, but it fills a logically distinct and previously unrecorded gap.

major comments (2)

[§3.2, Lemma 3.2] Lemma 3.2 asserts the unweighted shell sum S_m(h,q) ≪ (Y/√r + Y^{4/5} + √(Y r))(log Y)^4. The only justification offered is the classical weighted cumulative bound for ∑_{p≤Y} log p · e(αp), followed by the literal comparison “S_m(h,q) < ∑ log p e”. This step is invalid on three counts: (i) the classical bound is for the cumulative sum up to Y, not the short shell T_m = [10^{m-1},10^m); (ii) an unweighted sum is not dominated by a weighted sum without partial summation (or a uniform lower bound on log p together with control of the error); (iii) the passage from the full sum to the shell difference is never written. Because Lemma 3.3 feeds this majorant directly into Erdős–Turán, and Lemma 4.1 and Theorem 2.2 rest on Lemma 3.3, the gap is load-bearing. A correct write-up via partial summation plus differencing of two cumulative sums (or a short-interval form of the Vinogradov estimate) i
[§3.3 and Lemma 3.3] Even after the weighted-to-unweighted transfer is repaired, the range of the Dirichlet denominator r = q/gcd(h,q) ≥ (log Y)^{2A+8} is used to absorb the three terms of the majorant into Y/(log Y)^{A+4}. The argument assumes the same majorant constants remain valid uniformly for every q = 10^{k+1} with C_1 log m ≤ k ≤ m − C_2 log m. This uniformity should be stated explicitly (or a reference to a short-interval version of the bound supplied), since the subsequent choice of H and the admissible range of k depend on it.

minor comments (4)

[§3.2] The comparison symbol “<” in the display of Lemma 3.2 is notationally incorrect even if the intended majorization held; replace by the proper ≪ after partial summation.
[References] References [18] and [19] are the authors’ own cryptanalysis papers and have no bearing on digit equidistribution of primes; they should be removed or replaced by relevant surveys.
[Title] The title’s phrase “The Prime Digit Distribution Conjecture: A Formal Proof /*……*/” is slightly misleading; the body carefully distinguishes the averaged statement from the open pointwise conjecture. A milder title would better match the actual theorem.
[§5] In the proof of Lemma 5.1 the constant 1/5 is arbitrary; any fixed ratio <1 works, but the text should note that the implied constant in N_n ≍ M_n depends on the choice.

Circularity Check

1 steps flagged · score 1.0 of 10

Mild non-load-bearing self-citation of authors' own crypto papers as 'literature surveys'; derivation itself uses independent classical tools (PNT, Erdős–Turán, Vinogradov) with no fitted parameters or definitional loops.

self citation load bearing [Introduction, paragraph discussing stronger claims vs. Mauduit–Rivat]
"As emphasized in recent literature surveys [18, 19], the stronger claim, that every decimal digit occurs with limiting frequency 1/10 once all digit positions of a prime are pooled together, remains, to date, an open problem rather than a theorem, and is logically distinct from the residue-class equidistribution of the sum of digits..."

References [18] and [19] are the present authors' own cryptanalysis papers (genetic algorithms for RSA factorization), not independent surveys. They are used to underwrite the claim that the averaged equidistribution result was previously open. The citation is not load-bearing for any step of the proof itself, which never invokes [18,19]; it is only contextual framing.

full rationale

The claimed derivation of averaged digit equidistribution (Theorem 2.2) proceeds from the Prime Number Theorem, the Erdős–Turán discrepancy inequality, and classical Vaughan–Vinogradov exponential-sum bounds (via Davenport), none of which encode or assume the target frequency 1/10. No parameters are fitted to prime digit counts; the interior-digit lemma, shell dilution, and geometric summation are assembled from those external estimates. The sole self-reference appears in the introduction, where the authors cite their own RSA/genetic-algorithm papers [18,19] as 'recent literature surveys' asserting that pooled digit equidistribution remained open; this is purely contextual framing and is not invoked inside any lemma or the proof of Theorem 2.2. Consequently the logical chain is self-contained against external classical benchmarks and exhibits no definitional circularity, fitted-input-as-prediction, uniqueness import, or ansatz smuggling. Score 1 reflects only the mild, non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The argument imports three standard analytic-number-theory black boxes and introduces no free parameters or new physical/mathematical entities. All error-term constants are absolute or depend only on the free truncation parameter A chosen by the authors; nothing is fitted to digit data.

assumptions (4)

standard math Prime Number Theorem: π(x) ∼ x/log x (used for shell sizes π_m, total mass N_n, and final error conversion).
Invoked in Lemma 5.1 and the proof of Theorem 2.2; unconditional and classical.
standard math Erdős–Turán discrepancy inequality relating interval counts to exponential sums (Lemma 3.1).
Cited from Montgomery’s CBMS lectures; standard discrepancy tool.
standard math Vaughan–Vinogradov bound on Σ Λ(n)e(αn) for α near a/r (Lemma 3.2, Davenport).
Classical exponential-sum estimate over primes; the paper’s application to the m-digit shell is the only non-routine step.
domain assumption Interior positions C_1(A) log m ≤ k ≤ m − C_2(A) log m are equidistributed once the exponential-sum bound holds (Lemma 3.3).
This is the paper’s principal reduction; it stands or falls with Lemma 3.2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Prime Digit Distribution Conjecture: A Formal Proof of Average Digit Equidistribution in the Prime Numbers." pith.science (2026). https://pith.science/paper/5WZDB6NC

@misc{pith2026260710654,
  author       = {Pith},
  title        = {Pith review of: The Prime Digit Distribution Conjecture: A Formal Proof of Average Digit Equidistribution in the Prime Numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5WZDB6NC}},
  note         = {Machine review of arXiv:2607.10654}
}

abstract

Let $S_n=\{p\in\mathbb{P}:p<10^n\}$, $N_n$ denote the total number of decimal digits occurring in the primes of $S_n$, $C_n(d)$ be the number of occurrences of a digit $d\in\{0,\ldots,9\}$ among those digits, and $P_n(d)$ be the probability of occurrence of a digit, $d$ among those digits. We prove that \[ P_n(d)=\frac{C_n(d)}{N_n} =\frac{1}{10} +O\!\left(\frac{\log n}{n}\right), \qquad n\to\infty, \] uniformly for every decimal digit $d$. The argument is entirely unconditional and combines the Prime Number Theorem, the Erd\H{o}s--Tur\'an discrepancy inequality, and classical Vaughan--Vinogradov estimates for exponential sums over primes. The principal step establishes quantitative equidistribution for interior digit positions, while the logarithmically many exceptional positions near the ends of the decimal expansion are shown to have asymptotically negligible influence after averaging over all digit positions and prime lengths. Consequently, the decimal digits occurring in primes, when pooled over all positions and all primes below $10^n$, become asymptotically equidistributed. We also clarify the precise scope of the theorem by distinguishing this averaged equidistribution result from the substantially stronger and presently unresolved questions concerning pointwise digit equidistribution, normality, and higher-order digit correlations in the sequence of prime numbers.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 2 linked inside Pith

[1]

Sur les nombres qui ont des propri´ et´ es additives et multiplicatives donn´ ees.Acta Arithmetica, 13(3):259–265, 1968

A Gelfond. Sur les nombres qui ont des propri´ et´ es additives et multiplicatives donn´ ees.Acta Arithmetica, 13(3):259–265, 1968

1968
[2]

On the joint distribution ofq-additive functions in residue classes.Journal of Number Theory, 74(2):307–336, 1999

Dong-Hyun Kim. On the joint distribution ofq-additive functions in residue classes.Journal of Number Theory, 74(2):307–336, 1999

1999
[3]

The sum-of-digits function of canonical number systems: Distribution in residue classes.Journal of Number Theory, 132(12):2756–2772, 2012

Manfred G Madritsch. The sum-of-digits function of canonical number systems: Distribution in residue classes.Journal of Number Theory, 132(12):2756–2772, 2012

2012
[4]

Sur un probl` eme de gelfond: la somme des chiffres des nombres premiers.Annals of Mathematics, pages 1591–1646, 2010

Christian Mauduit and Jo¨ el Rivat. Sur un probl` eme de gelfond: la somme des chiffres des nombres premiers.Annals of Mathematics, pages 1591–1646, 2010

2010
[5]

Primes with an average sum of digits

Michael Drmota, Christian Mauduit, and Jo¨ el Rivat. Primes with an average sum of digits. Compositio Mathematica, 145(2):271–292, 2009

2009
[6]

The sum-of-digits function of polynomial sequences.Journal of the London Mathematical Society, 84(1):81–102, 2011

Michael Drmota, Christian Mauduit, and Jo¨ el Rivat. The sum-of-digits function of polynomial sequences.Journal of the London Mathematical Society, 84(1):81–102, 2011

2011
[7]

Exponential sums over primes in an arithmetic progression

Antal Balog and Alberto Perelli. Exponential sums over primes in an arithmetic progression. Proceedings of the American Mathematical Society, 93(4):578–582, 1985

1985
[8]

On the correlation of the sum of digits along prime numbers.convergence, 6(1):3, 2021

Karam Aloui, Christian Mauduit, and Mohamed Mkaouar. On the correlation of the sum of digits along prime numbers.convergence, 6(1):3, 2021

2021

Show all 21 references

[9]

Prime numbers in two bases.Duke Mathematical Journal, 169(10):1809–1876, July 2020

Michael Drmota, Christian Mauduit, and Jo¨ el Rivat. Prime numbers in two bases.Duke Mathematical Journal, 169(10):1809–1876, July 2020

2020
[10]

Prime numbers along rudin–shapiro sequences.Journal of the European Mathematical Society, 17(10):2595–2642, 2015

Christian Mauduit and Jo¨ el Rivat. Prime numbers along rudin–shapiro sequences.Journal of the European Mathematical Society, 17(10):2595–2642, 2015

2015
[11]

Exponential sums with automatic sequences.arXiv preprint arXiv:1710.01091, 2017

Sary Drappeau and Clemens M¨ ullner. Exponential sums with automatic sequences.arXiv preprint arXiv:1710.01091, 2017

2017 arXiv
[12]

Digital functions along squares of prime numbers.arXiv preprint arXiv:2509.17474, 2025

Michael Drmota and Jo¨ el Rivat. Digital functions along squares of prime numbers.arXiv preprint arXiv:2509.17474, 2025

2025
[13]

American Mathematical Society, 2025

Michael Drmota, Clemens M¨ ullner, and Lukas Spiegelhofer.Primes as sums of Fibonacci numbers, volume 305. American Mathematical Society, 2025

2025
[14]

Primes with restricted digits.Inventiones mathematicae, 217(1):127–218, 2019

James Maynard. Primes with restricted digits.Inventiones mathematicae, 217(1):127–218, 2019

2019
[15]

Prescribing the binary digits of primes, ii.Israel Journal of Mathematics, 206(1):165–182, 2015

Jean Bourgain. Prescribing the binary digits of primes, ii.Israel Journal of Mathematics, 206(1):165–182, 2015

2015
[16]

Prime numbers with a positive proportion of preassigned digits.Proceedings of the London Mathematical Society, 121(1):83–151, 2020

Cathy Swaenepoel. Prime numbers with a positive proportion of preassigned digits.Proceedings of the London Mathematical Society, 121(1):83–151, 2020

2020
[17]

Primes with a missing digit: Distribution in arithmetic progressions and an application in sieve theory.Journal of the London Mathematical Society, 109(1):e12837, 2024

Kunjakanan Nath. Primes with a missing digit: Distribution in arithmetic progressions and an application in sieve theory.Journal of the London Mathematical Society, 109(1):e12837, 2024. 14

2024
[18]

Cryptanalysis of rsa cryptosystem: Prime factor- ization using genetic algorithm.arXiv preprint arXiv:2407.05944, 2024

Mahadee Al Mobin and Md Kamrujjaman. Cryptanalysis of rsa cryptosystem: Prime factor- ization using genetic algorithm.arXiv preprint arXiv:2407.05944, 2024

2024 arXiv
[19]

A sieve-driven genetic algorithm for rsa cryptosystem: Exploiting digit distribution and algebraic regularities.Applied Soft Computing, page 115415, 2026

Mahadee Al Mobin. A sieve-driven genetic algorithm for rsa cryptosystem: Exploiting digit distribution and algebraic regularities.Applied Soft Computing, page 115415, 2026

2026
[20]

Number 84 in CBMS Regional Conference Series in Mathematics

Hugh L Montgomery.Ten lectures on the interface between analytic number theory and har- monic analysis. Number 84 in CBMS Regional Conference Series in Mathematics. American Mathematical Soc., 1994

1994
[21]

Springer Science & Business Media, 2013

Harold Davenport.Multiplicative number theory. Springer Science & Business Media, 2013. 15

2013

Pith tools

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

[1] [1]

Sur les nombres qui ont des propri´ et´ es additives et multiplicatives donn´ ees.Acta Arithmetica, 13(3):259–265, 1968

A Gelfond. Sur les nombres qui ont des propri´ et´ es additives et multiplicatives donn´ ees.Acta Arithmetica, 13(3):259–265, 1968

1968

[2] [2]

On the joint distribution ofq-additive functions in residue classes.Journal of Number Theory, 74(2):307–336, 1999

Dong-Hyun Kim. On the joint distribution ofq-additive functions in residue classes.Journal of Number Theory, 74(2):307–336, 1999

1999

[3] [3]

The sum-of-digits function of canonical number systems: Distribution in residue classes.Journal of Number Theory, 132(12):2756–2772, 2012

Manfred G Madritsch. The sum-of-digits function of canonical number systems: Distribution in residue classes.Journal of Number Theory, 132(12):2756–2772, 2012

2012

[4] [4]

Sur un probl` eme de gelfond: la somme des chiffres des nombres premiers.Annals of Mathematics, pages 1591–1646, 2010

Christian Mauduit and Jo¨ el Rivat. Sur un probl` eme de gelfond: la somme des chiffres des nombres premiers.Annals of Mathematics, pages 1591–1646, 2010

2010

[5] [5]

Primes with an average sum of digits

Michael Drmota, Christian Mauduit, and Jo¨ el Rivat. Primes with an average sum of digits. Compositio Mathematica, 145(2):271–292, 2009

2009

[6] [6]

The sum-of-digits function of polynomial sequences.Journal of the London Mathematical Society, 84(1):81–102, 2011

Michael Drmota, Christian Mauduit, and Jo¨ el Rivat. The sum-of-digits function of polynomial sequences.Journal of the London Mathematical Society, 84(1):81–102, 2011

2011

[7] [7]

Exponential sums over primes in an arithmetic progression

Antal Balog and Alberto Perelli. Exponential sums over primes in an arithmetic progression. Proceedings of the American Mathematical Society, 93(4):578–582, 1985

1985

[8] [8]

On the correlation of the sum of digits along prime numbers.convergence, 6(1):3, 2021

Karam Aloui, Christian Mauduit, and Mohamed Mkaouar. On the correlation of the sum of digits along prime numbers.convergence, 6(1):3, 2021

2021