REVIEW 3 minor 12 references
The Zsiflaw--Legeis theorem for arbitrary bases
T0 review · 0 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read This paper proves that for every base $g \ge 2$ the digital reverses of primes are equidistributed in arithmetic progressions, with a quantitative Siegel–Walfisz type error term.
desk verdict Strong paper: new weakly digital function framework, clean proof of reversed-prime equidistribution for every base, though the main theorem is independently in Dartyge-Rivat-Swaenepoel; send to referees. 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 machinery is the notion of a weakly digital function: $f_\lambda(n) = \sum_{0 \le i < \lambda} \alpha_i(\varepsilon_i(n))$, where each digit position $i$ has its own weight function $\alpha_i$ rather than a single common $\alpha$. This generalizes the Martin–Mauduit–Rivat digital functions just enough to cover the digital reverse, which grows like $g^{\mathrm{len}(n)}$ and is therefore not a classical digital function. The proof builds normalized exponential sums $F_\lambda^{[j]}(\beta)$, their product formula, and pointwise ($L^\infty$), discrete $L^1$, and hybrid bounds; the central quantity is $\sigma_\lambda(\alpha) = \sum_{i<\lambda} \gamma_i(\alpha)$, a sum of local gains coming from second differences of the positions-weighted digit map. For the reverse, applying the seed $\alpha_{L,i}(n) = \alpha n g^{L-i-1}$ and Lemma 26 turns any nonzero lower bound on $\min_i \|g^i(g^2-1)\alpha\|$ into the logarithmic growth $\sigma_\lambda \gg \lambda / \log(1/\sigma)$, which is what converts the general exponential-sum theorem into the Siegel–Walfisz type error term.
What would settle it
Take a small admissible case, say $g=2$, $q=3$, $a=1$, and directly evaluate the sum $\sum_{L-\lambda \le i < L} \|g^i(g^2-1)\alpha\|^2$ in Lemma 26 for $\alpha = h/q$ with a few $h$; the claimed lower bound is $\gg \lambda / \log(1/\sigma) + O(1)$ for every admissible $h$ and every $\lambda \le L$. A single admissible triple for which this quantity stays bounded while $\lambda$ grows, or a numerical comparison of $\pi_L^-(a,q)$ against the main term in Theorem 2 for such $g$ and $q$, would settle the claim.
Extended reading notes
Core claim
On the paper's own terms, the central claim is Theorem 3: for $g \ge 2$, $L \in \mathbb{N}$ and $2 \le x \le g^L$, the exponential sum $S = \sum_{n \le x} \Lambda(n) e(f_L(n))$ satisfies $S \ll x g^{-\kappa} (\log x)^4$, where $\kappa = \frac{1}{10} \sigma_\xi(\alpha)$ and $\xi = \lfloor \tfrac{1}{4} \log x / \log g \rfloor$. Theorem 2 then gives the quantitative count of reversed primes of length $L$: writing $\pi_L^-(a,q)$ for the number of $p \in [g^{L-1}, g^L)$ with $\mathrm{rev}(p) \equiv a \pmod q$, one has $\pi_L^-(a,q) = \frac{\rho_g(a,q)}{q} \frac{g^L}{\log g^L}(1 + O(1/L)) + O(g^L \exp(-c \sqrt{L}))$ whenever $q \le \exp(c \sqrt{L})$ and $(a,q,g^2-1)=1$ with $g \nmid (a,q)$. The explicit factor $\rho_g(a,q)$ encodes the necessary coprimality conditions, and the error term is effective, depending only on $g$.
Load-bearing premise
The quantitative gain relies on Lemma 26, which assumes that $\sigma = \min_{0 \le i \le L} \|g^i(g^2-1)\alpha\| > 0$ and then proves $\sigma_\lambda(\alpha_L) \gg \lambda / \log(1/\sigma) + O(1)$; for $\alpha = h/q$ this is exactly the condition $q \nmid g^L(g^2-1)h$. If that lower bound ever failed for some base and modulus, the error term would no longer decay as $\exp(-c \log x / \log q)$.
Editorial extensions
If this is right
- Theorem 1: for every base $g \ge 2$ there are infinitely many primes $p$ with $\mathrm{rev}(p) \equiv a \pmod q$ whenever $(a,q,g^2-1)=1$ and $g \nmid (a,q)$.
- Theorem 2 gives the asymptotic count for length-$L$ reversed primes with relative error $O(1/L)$ plus an exponentially small term, uniformly for moduli up to $\exp(c\sqrt{L})$.
- Theorem 5 extends the equidistribution statement from fixed length $L$ to the absolute reverse $\mathrm{rev}(p)$ summed up to $x$, with error $x \exp(-c \log x/\log(q+1))$ under $q \le \exp(c \log x/\log\log x)$.
- The proof upgrades the previous base restriction ($g \ge 31699$, later $26000$) to all $g \ge 2$, and the paper notes that the same result was obtained independently by Dartyge–Rivat–Swaenepoel.
Reading between the lines
- The weakly digital framework is not tied to the reverse map; the same $L^\infty$/$L^1$/hybrid bounds should give equidistribution results for other position-dependent digit functions, such as partial reverses or digit permutations, provided an analogue of Lemma 26 holds.
- The error term $\exp(-c\sqrt{L})$ in Theorem 2 is likely far from sharp by analogy with Siegel–Walfisz; a sharper treatment of the minor-arc contribution could plausibly yield $\exp(-c L)$ or $L^{-C}$.
- A direct numerical comparison of the two independent proofs (this paper and [4]) could expose which auxiliary bound dominates the admissible modulus range, guiding further refinements.
- One testable extension is to count reversed primes in short intervals of the form $[x, x+x^{1/2+\epsilon}]$; the Type I/II bounds here are tailored to the full range up to $x$, and the short-interval version would measure how much of the method survives.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves analogues of Dirichlet's theorem and the Siegel--Walfisz theorem for digital reverses of primes in arbitrary base g >= 2. The authors introduce a class of "weakly digital functions" f_{lambda,alpha} whose digit maps may depend on position, and establish a quantitative exponential sum estimate over primes (Theorem 3) with decay g^{-kappa}. The key innovation is a lower bound for the digit-coherence parameter sigma_lambda(alpha_L) for the linear seed alpha_L,i(n)=alpha n g^{L-i-1}, which is then applied to alpha=h/q. This yields a splitting of frequencies in Theorem 4 according to whether q divides g^L(g^2-1)h, producing the main term (q,g^L(g^2-1))/q times a sharp count plus a strong error term. Theorem 5 converts the relative digital reverse to the absolute reverse, and Theorems 1 and 2 follow from the argument of Section 11 of the authors' previous paper [1]. The proof follows the Vaughan--van der Corput--Gallagher--Sobolev route, with explicit parameter choices z=x^{1/4}, theta=1/4, and R=g^{2kappa_II}; the paper is self-contained for all exponential sum estimates.
Significance. If correct, this is a significant result: it removes the previous base restriction (g >= 31699 in [1], improved to g >= 26000 in [2]) and establishes equidistribution of reversed primes in arithmetic progressions for every base g >= 2, with effective constants. The main term is derived rather than assumed, the exponential sum bound is unconditional and uniform in the seed alpha, and there are no fitted parameters. The proof contains clearly written Type I and Type II estimates with explicit parameter choices, and the delicate lower bound in Lemma 26 is the load-bearing step that makes the error term exp(-c log x / log q) possible. The paper also transparently acknowledges the independent work of Dartyge--Rivat--Swaenepoel [4].
minor comments (3)
- [Section 6, Eq. (6.22)] The displayed bound "S_II << M N R^2 x^{-5 kappa_II} log x + M N R^{-1/2}" does not match the preceding derivation: combining (6.11) with the bound on S_II(r) just above gives M N R^{3/2} g^{-5 kappa_II} log x + M N R^{-1/2}. After substituting R = g^{2 kappa_II}, both forms yield a final bound O(x g^{-kappa_II} log x), so the central estimate is unaffected, but the displayed formula should be corrected.
- [Lemma 26] The statement writes sigma_lambda(alpha_L) >> lambda/log(1/sigma) + O(1), which is misleading as written: for sigma not extremely small the right-hand side can exceed the universal upper bound sigma_lambda <= lambda/20 from (4.5). The proof actually gives sigma_lambda(alpha_L) >> lambda / (1 + (log(g/((g+1)sigma)))/log g) + O(1), i.e. a bound of the form >> lambda/(log(1/sigma)+O(1)). Since the application only uses sigma = 1/q, the main arguments are unaffected, but the lemma statement should be reformulated.
- [Section 8, Theorem 2 and Theorem 1] The final step deriving Theorems 1 and 2 from Theorem 5 is delegated to Section 11 of [1] without a sketch. Because [1] is a preprint and the conditions on (a,q,g^2-1) and g | (a,q) enter through rho_g(a,q), a brief indication of how the main term arises would improve self-containedness.
Circularity Check
No significant circularity: the main-term formula is derived from exact orthogonality and the exponential-sum gain is a genuine lower bound, not a fitted or self-imported input.
full rationale
The derivation is self-contained in the relevant sense. Theorem 3 is an unconditional bound for arbitrary weakly digital functions, with the gain kappa defined through sigma_xi(alpha); in the application to alpha=h/q, Lemma 26 proves a genuine lower bound from the distance sigma = min_i ||g^i(g^2-1)alpha||. The main term in Theorem 4 is obtained by exact character orthogonality for frequencies satisfying q | g^L(g^2-1)h, and the complementary frequencies are bounded by the exponential-sum estimate; the density rho_g(a,q)/q is not fitted or assumed. No parameter is fitted to the equidistribution statement, and no uniqueness assertion or ansatz is imported from the authors' earlier work to force the choice. The final appeal to the argument of Section 11 of [1] is a citation to a proof method in the authors' previous paper, not an appeal to a theorem whose statement already contains the present conclusion; even if viewed as a self-citation, it is not load-bearing for the quantitative exponential-sum bound or the main-term formula. Hence there is no circular step to report.
Assumptions & free parameters
assumptions (5)
- standard math Vaughan's identity decomposes Lambda(n) into four bilinear terms (Lemma 24).
- standard math Van der Corput's inequality (Lemma 20, from Graham-Kolesnik).
- standard math Gallagher-Sobolev inequality (Lemma 16, from Montgomery [11]).
- standard math Dartyge-Martin-Rivat-Shparlinski-Swaenepoel Lemma 2.7 on lower bounds for ||g^i alpha|| (Lemma 25).
- domain assumption The digital reverse rev_L(n) is represented by a weakly digital function via alpha_{L,i}(n) = alpha n g^{L-i-1}.
invented entities (1)
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Weakly digital functions f_lambda,alpha generated by a sequence alpha_i of digit maps
Cite this review
Pith. "Pith review of The Zsiflaw--Legeis theorem for arbitrary bases." pith.science (2026). https://pith.science/paper/HYALLRE6
@misc{pith2026250708714,
author = {Pith},
title = {Pith review of: The Zsiflaw--Legeis theorem for arbitrary bases},
year = {2026},
howpublished = {\url{https://pith.science/paper/HYALLRE6}},
note = {Machine review of arXiv:2507.08714}
}
read the original abstract
In this paper, we prove analogues of the Dirichlet theorem on arithmetic progressions and the Siegel--Walfisz theorem for the digital reverses of primes for arbitrary bases, which the authors obtained in the previous paper but only for large bases. The proof is based on a generalization of the result of Martin--Mauduit--Rivat (2014) on the exponential sums over primes with the so-called ``digital'' functions.
Reference graph
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Reviewed August 6, 2026 · model on record in the stance chip above.
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