{"id":"9818dfbd-90d5-461b-93f9-7a9507715561","arxiv_id":"2507.08714","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The digital reverse of primes is equidistributed in arithmetic progressions for every base g>=2, with a quantitative error term.","lead":"This paper proves that, in every number base, the reversed-digit form of prime numbers is equidistributed across allowed arithmetic progressions, with a Siegel-Walfisz style error term. Its new technique, weakly digital functions, may be reusable for other digit-based prime problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the delicate Lemma 26 lower bound and the Theorem 4 frequency split both check out; no load-bearing flaw found.","rationale":"The reader's verdict of ACCEPT with moderate confidence is appropriate. I attempted to locate a concrete failure in the one place where the entire quantitative gain originates, namely Lemma 26, and in the way Theorem 4 forces sigma >= 1/q in the complementary frequencies. Both steps are valid. Using the pair (0,1) in (4.2) is legitimate because gamma_i is a sum over all pairs, so a single pair provides a lower bound; the reduction from exponent L-i-2 to L-i-1 only costs a g-dependent constant. The exactness of the main-term split is elementary: for alpha = h/q, sigma = 0 iff q divides g^L(g^2-1)h, and otherwise every relevant distance is at least 1/q. The rest of the machinery, including Lemma 17, the Type I and Type II estimates, and the Vaughan decomposition, is a faithful adaptation of the Martin-Mauduit-Rivat framework and is internally coherent. The constants can be tuned so the (log x)^4 factor is absorbed under the stated q <= exp(c log x / log log x) condition. The only minor issue is a notational slip in (6.22), which does not affect the final bound after the choice R = g^{2 kappa}. There is no machine-checked formalization, so residual risk lies in the intricate bookkeeping rather than in any identified false step.","tokens_in":29528,"tokens_out":47077,"duration_ms":521416,"concrete_test":"One check that would settle the delicate step: for g in {2,3,5}, q <= 200, 1 <= h < q with gcd(h,q)=1 and q not dividing g^L(g^2-1)h, and L=16,32,64, compute sigma_lambda(alpha_L) at lambda = floor((1/4) log_g(g^L)) using the definition (4.2) and compare with C_g * lambda / (log_2 q + 1). A single violation would break the saving in Theorem 4; agreement for all small cases would independently confirm that the (0,1) contribution and the Lemma 25 block argument are correctly quantified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The proof's least obvious step is the Lemma 26 lower bound and the associated exact split in Theorem 4; I checked that step rather than stopping at the face of the statement. For alpha_{L,i}(n)=alpha n g^{L-i-1}, the (m,n)=(0,1) contribution to gamma_i is ||g^{L-i-2}(g^2-1)alpha||^2, which is >> ||g^{L-i-1}(g^2-1)alpha||^2, with the constant g^{-2} absorbed into the implied constant. The block argument partitions [L-lambda,L) into J << 1 + log(1/sigma)/log 2 blocks and applies Lemma 25 to get a contribution >= (g+1)^{-2} in each block, so sigma_lambda(alpha_L) >> lambda/log(1/sigma) is sound. In Theorem 4 the complementary frequencies are exactly those with sigma >= 1/q, because if q does not divide g^L(g^2-1)h then q cannot divide g^i(g^2-1)h for any 0 <= i <= L. The Type I and Type II estimates, including the eta_g bookkeeping around (4.14) and the monotonicity in Lemma 18, are internally consistent. The only blemish I found is cosmetic: (6.22) displays R^2 x^{-5 kappa} where the surrounding calculation suggests R^2 g^{-5 kappa}; after choosing R = g^{2 kappa} both forms are harmless, so the central bound is unaffected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":29795,"tokens_out":33748,"duration_ms":346834,"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].","major_comments":[],"minor_comments":[{"comment":"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.","section":"Section 6, Eq. (6.22)"},{"comment":"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":"Lemma 26"},{"comment":"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.","section":"Section 8, Theorem 2 and Theorem 1"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound on the points I checked. The only issues are local presentation problems: a harmless but incorrect displayed formula in (6.22) and an imprecise statement of Lemma 26. The reliance on Section 11 of the previous preprint for the final theorems is acceptable for a sequel, though a short outline would make the paper more self-contained. The overlap with [4] is acknowledged in the introduction, so there is no novelty concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the main theorem — Dirichlet and Siegel-Walfisz analogues for reversed primes for every base g>=2 — was independently proved by Dartyge-Rivat-Swaenepoel [4, Thm 1.4] during final write-up, and the paper says so in a Remark. So the central result is not new to the literature. Second, what is new is the weakly digital function framework (Definition 1) and the proof route; this is a real methodological contribution, not a repackaging.\n\nThe authors generalize Martin-Mauduit-Rivat's digital functions to allow the digit map to depend on position, proving an exponential sum bound for primes weighted by such functions (Theorem 3). The application to reversed primes then goes through cleanly. The proof follows the standard Vaughan / Van der Corput / Gallagher-Sobolev machinery, with explicit parameter choices (z=x^{1/4}, theta=1/4, R=g^{2 kappa_II}). The Type I and Type II estimates are written out in detail. The delicate lower bound in Lemma 26 — the positivity of sigma = min_i ||g^i(g^2-1) alpha|| — is the least obvious step; I checked the block argument and the split in Theorem 4, and it holds up. There is a cosmetic typo at (6.22): R^2 x^{-5 kappa} should be R^2 g^{-5 kappa}, harmless after choosing R = g^{2 kappa}.\n\nSoft spots: the main theorems are not new relative to [4], so a referee should calibrate novelty accordingly; this paper's contribution is the framework and the self-contained proof. The proof is intricate and is not machine-checked, but I found no load-bearing contradiction. The Remark is honest disclosure, and the dependence on [4] is confined to the result being independent; the methods differ.\n\nWho this is for: analytic number theorists who work on digital functions and primes in arithmetic progressions. It deserves a serious referee; I would send it out rather than desk-reject, and I'd probably accept after minor revisions (fixing the typo and maybe sharpening the novelty discussion).","headline":"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.","tokens_in":30403,"tokens_out":2800,"would_cite":true,"duration_ms":27690,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A63","11N05","11N69"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["reversed primes","digital reverse","weakly digital functions","arithmetic progressions","Siegel–Walfisz theorem","exponential sums over primes","base-g representation","Dirichlet theorem for primes"],"falsifier":"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.","tokens_in":29256,"feed_emoji":"🔁","tokens_out":7056,"duration_ms":66897,"temperature":0.7,"pith_summary":"The paper proves that the digital reverses of primes—integers obtained by reading a prime's base-$g$ digits backwards—are equidistributed in arithmetic progressions for every base $g \\ge 2$. Earlier work by the same authors had this only for large bases ($g \\ge 31699$, later improved to $26000$); this paper removes the constraint entirely. If correct, the reversed primes satisfy analogues of Dirichlet's theorem and the Siegel–Walfisz theorem with a quantitative error term, for every base. The engine is a new class of \"weakly digital functions\" that lets the authors carry over Mauduit–Rivat style exponential-sum bounds to functions like the digital reverse, which are too large to be handled by the classical digital-function framework.","feed_headline":"Reversed primes spread evenly in every number base","feed_subtitle":"A new proof covers bases as small as 2, with an effective Siegel–Walfisz type error term.","key_machinery":"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.","core_discovery":"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 $g$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the general exponential-sum framework for digital functions that the paper adapts to weakly digital functions.","marker":"[7]"},{"why":"Provides the Mauduit–Rivat method for exponential sums over primes with digit functions, the direct ancestor of the proof.","marker":"[8]"},{"why":"The authors' previous paper, which proved the same theorems for bases at least 31699 and supplies the reduction from Theorem 5 to Theorems 1 and 2.","marker":"[1]"},{"why":"The independent simultaneous result by Dartyge, Rivat and Swaenepoel, noted in the paper as having obtained the same theorems.","marker":"[4]"},{"why":"Dartyge–Martin–Rivat–Shparlinski–Swaenepoel; Lemma 25 is cited from it, giving the lower bound $\\|g^{i_0}\\alpha\\| \\ge 1/(g+1)$ used in Lemma 26.","marker":"[3]"},{"why":"Montgomery's Topics in Multiplicative Number Theory is cited for the Gallagher–Sobolev inequality used to pass from $L^1$ bounds to sums over rationals.","marker":"[11]"},{"why":"Graham–Kolesnik supplies the Van der Corput inequality used in the Type II estimate.","marker":"[6]"}],"fun_headline_variants":["Reversed primes uniform in all bases, even binary","All bases now: reversed primes obey Siegel-Walfisz","Binary included: precise count of reversed primes","No base too small: reversed primes equidistributed","Effective error term: reversed primes for any base"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $\\exp(-c \\log x / \\log q)$.","fun_headline_variants_meta":{"raw":{"variants":["Reversed primes uniform in all bases, even binary","All bases now: reversed primes obey Siegel-Walfisz","Binary included: precise count of reversed primes","No base too small: reversed primes equidistributed","Effective error term: reversed primes for any base"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000483,"raw_usage":{"total_tokens":2362,"prompt_tokens":900,"completion_tokens":1462,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":1388}},"tokens_in":516,"tokens_out":1462,"duration_ms":11837,"temperature":1.0,"reasoning_tokens":1388,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:12:24.580533+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Martin, C","cited_arxiv_id":null,"evidence_quote":"Supplies the general exponential-sum framework for digital functions that the paper adapts to weakly digital functions."},{"cited_title":"Mauduit and J","cited_arxiv_id":null,"evidence_quote":"Provides the Mauduit–Rivat method for exponential sums over primes with digit functions, the direct ancestor of the proof."},{"cited_title":"Dartyge and B","cited_arxiv_id":null,"evidence_quote":"Dartyge–Martin–Rivat–Shparlinski–Swaenepoel; Lemma 25 is cited from it, giving the lower bound $\\|g^{i_0}\\alpha\\| \\ge 1/(g+1)$ used in Lemma 26."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Montgomery's Topics in Multiplicative Number Theory is cited for the Gallagher–Sobolev inequality used to pass from $L^1$ bounds to sums over rationals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Graham–Kolesnik supplies the Van der Corput inequality used in the Type II estimate."}],"review_version":1}