{"id":"7e7d03b2-01ae-4ebe-b718-24e585025df9","arxiv_id":"1908.06808","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A general matrix-inversion recursion computes Euler products over primes in lattice-invariant residue classes to hundreds of digits with explicit error bounds, yielding new high-precision values of Shank's and Lal's constants.","lead":"The paper gives a fast algorithm, with rigorous error bounds, for computing certain Euler products over primes in residue classes modulo q, and uses it to produce hundreds of digits of several number-theoretic constants. The method extends Shanks's classical scheme and also applies to products of rational functions of primes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Error bound in Thm 1.3 grows with r for q=5,8,9 (A=|G7|d(expG)/2>2), so the stated bound does not prove the claimed double-exponential convergence.","rationale":"Good-faith reading: the paper's substantive contribution is a matrix recursion expressing logs of lattice-invariant Euler products in terms of easier 'Gamma' quantities, with an explicit remainder formula (Theorem 1.3) and an extension to rational Euler products (Theorem 1.4). The algebra appears sound: the generalized Moebius inversion (Proposition 4.1) is standard, the matrices N_d match the detailed modulo-7 example, and the numerical values are cross-validated against direct truncation and published constants. So the core identities are not in doubt. The load-bearing weakness is the gap between the 'double-exponential time' headline and the explicit bounds. Lemma 4.3 bounds the discarded tail V_s(D) by roughly 1/(P D), which hides the true exponential-in-D decay P^{-Ds} that makes Shanks-style iteration double-exponential. As a result, the Theorem 1.3 error bound contains (|G7|d(expG)/2)^{r-1}/(2^r), which grows with r for many small moduli such as q=5, 8, 9, and 11. For these q the theorem gives no way to choose r to reach a target precision; the only lever is P, which costs exponential time. This is a more concrete and more immediate version of the reader's concern about |G7|: even if |G7| were bounded by q^ε, the stated bounds still do not deliver the advertised complexity. Section 7's explicit admission that |G7| is not shown to be q^ε is a second, independent obstacle to uniformity in q. The reader's verdict of CONDITIONAL remains appropriate: the algebra and numerics are credible, but the paper must either prove and state a sharp tail bound for V_s(D) and demonstrate the double-exponential decrease, or substantially rewrite the complexity claim to match the bounds actually proved, and make the Sage script accessible so the numerical claims are reproducible. My concern does not move the verdict; it sharpens the reasons for the condition.","tokens_in":24457,"tokens_out":24875,"duration_ms":251614,"concrete_test":"Evaluate the right-hand side of Theorem 1.3 for q=5, s=2, P=100, and r=2,3,4,5 using the Lemma 4.3 tail bound. With |G7|=3 and d(expG)=d(4)=3, the bound is (1/2)(1+(r-1)/3)(9/2)^{r-1}(1+100/(2·2^r-1))/(200·2^r); it increases from about 5.7×10^{-2} at r=2 to about 1.5×10^{-1} at r=5, far above the 100-digit accuracy claimed in Section 9. This shows Theorem 1.3 as stated cannot certify those digits. Then repeat the same computation with the sharp tail estimate ||V_s(D)|| ≤ P^{1-Ds}/(Ds-1) for D≥2^r; if the resulting error bound decays like P^{1-2^r s}·(9/2)^{r-1}, the double-exponential claim is recoverable, identifying the missing lemma needed to support the paper's central complexity assertion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract and Theorem 1.2 claim that lattice-invariant Euler products can be computed in double-exponential time, but the explicit error bound in Theorem 1.3 does not support this. For fixed q, the Theorem 1.3 bound after r iterations is essentially ((|G7|d(expG)/2)^{r-1})/(P s 2^r), using Lemma 4.3's tail estimate ||V_s(D)|| ≤ (1+P/(D-1))/(P D) ≈ 1/(P D). For q=5, |G7|=3 and d(expG)=3, so the base is 9/2=4.5 and the factor (4.5/2)^r = 2.25^r increases with r; for q=8, |G7|=4, expG=2, so the factor is 4^{r-1}/2^r = 2^{r-2}, also increasing. Therefore increasing the recursion depth r cannot drive the error bound to zero; the only remaining lever is P, and taking P ≈ 10^n to obtain n digits costs Ω(P/log P) operations, which is exponential in n, not double-exponential. The advertised double-exponential speed would require a sharper tail estimate such as ||V_s(D)|| ≈ P^{1-Ds}/(Ds-1), which the paper neither states nor proves. Section 7's admission that |G7| is not known to be q^ε compounds the issue: even with a sharp tail bound, the factor |G7|^{r-1} in the error could spoil uniformity in q. The worked modulo-7 example and numerical cross-checks suggest the algebraic recursion is correct, but the headline complexity assertion does not follow from the theorems as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines, for each modulus q, a family of lattice-invariant classes of (Z/qZ)^× and derives a recursive scheme for computing the logarithms of Euler products over primes in those classes. The main quantitative statement, Theorem 1.3, approximates the vector V_s(1) by a finite combination of Gamma_s(d_1...d_v) values, with an explicit error bound involving P, s, r, |G7|, and d(exp G). Theorem 1.4 extends the scheme to products of F(1/p)/G(1/p) for coprime real polynomials. The paper also supplies a Sage script, many high-precision numerical values, and cross-checks against direct truncation and known constants. The headline claim is that these Euler products can be computed in double-exponential time.","tokens_in":24787,"tokens_out":4994,"duration_ms":48069,"significance":"If the recursion and the associated error bounds were fully established, this would be a valuable algorithmic tool in analytic number theory, enabling high-precision evaluation of Euler products that occur in density problems. The algebraic core is appealing: Möbius inversion on the subgroup lattice, explicit matrices N_d, and a worked example modulo 7 make the recursion transparent. The provision of a Sage script and the extensive numerical cross-checks (against direct truncation, Finch's values, and Gourdon–Sebah) are concrete strengths that support the correctness of the identities themselves. However, the advertised double-exponential complexity is not supported by the theorem as stated, and one key lemma in the error analysis is left unproved.","major_comments":[{"comment":"The error bound in Eq. (12) does not decrease with r for many moduli, so it does not imply the double-exponential convergence claimed in Theorem 1.2 and the abstract. For q=5, one has |G7|=3 and d(exp G)=d(4)=3, so the base (|G7|d(exp G)/2) equals 9/2, and the r-dependent factor becomes (9/2)^{r-1}/2^r = (9/4)^{r-1}/2, which grows with r. For q=8, |G7|=4 and d(exp G)=2, giving (4)^{r-1}/2^r = 2^{r-2}, also increasing. Since Lemma 4.3 only provides ||V_s(D)|| <= (1+P/(D-1))/(P D) ≈ 1/(P D) for D ≥ 2^r, increasing r cannot drive the bound to zero for these moduli. The only remaining lever is P, and choosing P ≈ 10^n to obtain n digits costs Ω(P/log P) arithmetic operations, which is exponential in n, not double-exponential. A sharper tail estimate or a spectral analysis of N_2 is needed to substantiate the complexity claim.","section":"Theorem 1.3, Eq. (12); Section 1, Theorem 1.2"},{"comment":"Lemma 4.2 states that the coefficients of a product N_{d1}...N_{dv} are bounded by |G7|^{v-1} in absolute value, but no proof is provided in the text; the passage moves directly from the lemma statement to 'End of the proof of Theorem 1.3'. This bound is load-bearing, as it is used to derive the error term in Eq. (12). The authors should supply a proof or a precise reference for this estimate.","section":"Section 4, Lemma 4.2"},{"comment":"The authors state in Section 7 that they have not been able to establish |G7| <<_ε q^ε. This admission is directly relevant to Theorem 1.2, which claims double-exponential time 'for every modulus q ≥ 3'. The constants in Theorems 1.3 and 1.4 depend on |G7| and d(exp G); if |G7| grows faster than any power of q, the number of cyclic subgroups, the matrix sizes, and the iteration depth needed for a fixed precision could all grow too quickly. The paper should either prove a bound on |G7| or explicitly formulate the complexity statement with the q-dependence shown, rather than asserting a uniform double-exponential bound.","section":"Section 7, Proposition 7.3 and following paragraph"}],"minor_comments":[{"comment":"The sentence 'where d_1,...,d_r are all divisors of exp G excluding 1' is imprecise: the sums are over tuples (d_1,...,d_v) with v ≤ r-1 and with the product d_1...d_v restricted by d_1...d_v ≤ 2^r. Please rephrase to describe the actual summation range.","section":"Theorem 1.3 statement"},{"comment":"The quantity γ_s(G_0,t) is defined using L_P(ts,χ), so it depends on P, but this dependence is suppressed without comment. Please either write γ_{s,P} or state explicitly in the notation that P is fixed throughout.","section":"Equation (6) and surrounding text"},{"comment":"The displayed inequality for |log I_3| has unbalanced parentheses in the published text, making the final bound hard to read. Please check and correct the parenthesis structure.","section":"Theorem 1.4, bound (39)"},{"comment":"In the table of results, the line for the class 2,6,7,11 mod 13 reads 'p ≡ 2,6,7,11 r5s', which is a typo: it should read 'mod 13'. Please fix.","section":"Section 9, Modulo 13 row"},{"comment":"The phrase 'double-exponential time' is used without a definition or a model of computation. Please state precisely what quantity is counted (bit operations, evaluations of Γ, etc.) and with respect to which parameter the time is double-exponential.","section":"Abstract and Section 1"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a genuinely useful algebraic recursion and impressive numerical output, but the central complexity claim is not justified by the stated theorems. In my view this is fixable: the authors should either prove a sharper tail bound or matrix-norm estimate that makes the error decrease with r, or restrict Theorem 1.2 to moduli where the relevant constant is < 2 and otherwise state the complexity as exponential in n with a q-dependent constant. The omitted proof of Lemma 4.2 must be supplied. I would not reject, but the revision needs to be substantive rather than cosmetic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The method is real: a matrix recursion over cyclic subgroups of (Z/qZ)^× that generalizes Shanks and Moree–Osburn, plus a rational-function extension, and it produces 100+ digit values for Shank's and Lal's constants that check out against independent sources. The second thing is that the central error bound as printed is wrong. Lemma 4.3 states |log ζ| ≤ (1+P/(f-1))/(P f), but the proof in the same lemma derives P^{-f}(1+P/(f-1)). For f=Ds with D=2^r and P large, the printed denominator gives something like 1/(P s 2^r), so the error bound in Theorem 1.3 has an r-dependent factor (|G7| d(expG)/2)^{r-1}/(P s 2^r). For q=5, |G7|=3, d(expG)=3, that base is 9/4=2.25, so the bound grows with r; q=8 and 9 are similar. The theorem as written does not imply double-exponential convergence, or even convergence as r increases. But the proof of Lemma 4.3 contains the sharper estimate, so this looks like a typo that got propagated. Fixing it would give error ~ (|G7| d(expG)/2)^{r-1}/P^{s 2^r}, which is double-exponentially small in r for fixed q. That matters: the advertised complexity claim depends on it.\n\nWhat's good: the recursion itself, the explicit inverse (Prop 4.1), the worked example modulo 7, and the cross-checks against truncation and published values. The paper is also honest about not proving |G7| << q^ε, which is a real limitation for large q.\n\nOther soft spots: \"double-exponential time\" is imprecise—with the corrected bound it's double-exponential convergence, and the time for fixed q looks polynomial in the number of digits. The Sage script is referenced but no URL or checksum is given. Section 8 says precision is 'roughly handled' then claims displayed digits are always accurate; that needs clarifying.\n\nWho's this for: computational and analytic number theorists who need certified high-precision values for Euler products and want to see the Shanks idea systematized. It deserves a serious referee. My recommendation: send it out, but require the authors to fix Lemma 4.3 and Theorem 1.3, state the complexity claim precisely, and make the script accessible.","headline":"A genuinely useful computational method for Euler products, with a real but likely fixable error-bound typo that currently undermines the advertised double-exponential convergence.","tokens_in":25367,"tokens_out":10897,"would_cite":true,"duration_ms":99814,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11Y60","11N13","05A"],"pacs":[],"model":"deepseek-v4-flash","headline":"The authors prove Euler products over lattice-invariant classes can be computed in double-exponential time, with explicit error bounds.","keywords":["Euler products","lattice-invariant classes","Loeschian numbers","Lal's constant","multi-precision computation","Moebius inversion","Witt decomposition","double-exponential time"],"falsifier":"For $q$ equal to the product of many small primes, count the cyclic subgroups of $(\\mathbb{Z}/q\\mathbb{Z})^\\times$; a growth faster than any fixed power of $q$ would invalidate the claimed double-exponential speed.","tokens_in":1816,"feed_emoji":"🧮","tokens_out":3152,"duration_ms":142358,"temperature":0.7,"pith_summary":"This paper claims a recursive identity scheme that computes certain infinite products over primes in specified residue classes modulo q to hundreds of digits, with certified error bounds. The products covered are the lattice-invariant classes: primes whose residue generates a fixed cyclic subgroup of (Z/qZ)^×. Each step of the recursion replaces a product evaluated at exponent s by companion products at ds, pushing the argument to large exponents where the tail is negligible. The same machinery handles products of rational functions F(1/p)/G(1/p), which gives high-precision values of the Loeschian density constants and the classical constants treated in the paper.","feed_headline":"Euler product constants computed to 100+ digits fast","feed_subtitle":"Iterating the exponent s to 2s yields certified high-precision values for Loeschian and classical constants.","key_machinery":"The central object is a lattice-invariant class $A(K)$: the set of elements of $(\\mathbb{Z}/q\\mathbb{Z})^\\times$ that generate one fixed cyclic subgroup $K$. The partition of the unit group into these classes is exactly the coarsest partition that the character identities can distinguish. The workhorse is Lemma 2.2, which expands a product of $L_P(ts,\\chi)$ over characters trivial on a subgroup $G_0$ into a product over intermediate subgroups $K$ of factors $(1-p^{-|K/G_0|s})^{-|G/K|}$; this converts a single product at $s$ into companion products at $|K/G_0|s$. In matrix form this is $\\Gamma(t)=M_1 V_s(t)+\\sum_{d>1} M_d V_s(dt)$, and the paper inverts $M_1$ explicitly using the Möbius function of the subgroup lattice, whose value on $K\\subset H$ is the ordinary Möbius function of $|H/K|$. The resulting recursion $V_s(t)=M_1^{-1}\\Gamma(t)-\\sum_d (N_d/d)V_s(dt)$ is unfolded into the alternating sum of Theorem 1.3. For rational products, Lemma 6.1 expresses $F(t)/G(t)$ as $\\prod_{j\\ge2}(1-t^j)^{b_F(j)-b_G(j)}$, where the exponents come from the Newton-Girard recursion on the coefficients of the polynomials.","core_discovery":"At the center of the paper is Theorem 1.3: for every modulus $q \\ge 3$, every lattice-invariant class $A$, and every real $s>1$, the vector $V_s(1)$ of logarithms of the Euler products over the lattice-invariant classes can be approximated by an explicit finite alternating combination of the vectors $\\Gamma_s(d_1\\cdots d_v)$, with an explicit error bound of the form $\\tfrac12(1+\\tfrac{r-1}{|\\mathcal{G}_7|})(\\tfrac{|\\mathcal{G}_7| d(\\exp G)}{2})^{r-1}\\tfrac{1+P/(s2^r-1)}{P s 2^r}$. The $\\Gamma$ values on the right are computed from products of Dirichlet L-series at the same arguments, via Hurwitz zeta values, and the matrices $N_d$ are Möbius-weighted subgroup-jump matrices independent of $s$. Because the bound shrinks like $1/(s2^r)$, increasing $r$ gives multi-precision evaluation. Theorem 1.4 extends the same scheme to products $\\prod_{p\\in A} F(1/p)/G(1/p)$ with $F(0)=G(0)=1$ and $F-G$ divisible by $X^2$, using a Witt-type infinite factorization of $F(t)/G(t)$; the paper then derives the displayed 100-digit values of the two Loeschian density constants and the other constants in the introduction.","pith_inferences":["Not in the paper: the character identity of Lemma 2.2 is a finite Fourier inversion on the subgroup lattice, so fast zeta transforms over the divisor lattice of $\\exp G$ could reduce the cost of the sums over $d_1,\\dots,d_v$.","Not in the paper: because the main theorem is stated for the whole vector of lattice-invariant classes, the same certified recursion applies to arbitrary unions of classes by summing entries, so the precision guarantee should extend to any subset of residues modulo $q$ that is a union of classes.","Not in the paper: tabulating the number of cyclic subgroups for moduli with many prime factors would settle whether the $q^{\\epsilon}$ bound on $|\\mathcal{G}_7|$ is true; that numerical experiment would decide whether the practical range of the method is limited to small moduli or extends to all $q$."],"forward_implications":["For any lattice-invariant class modulo $q$ and any $s>1$, the Euler product $\\zeta_P(s;q,A)$ is computable in double-exponential time, with an accompanying computer-algebra script that reproduces the listed 100-digit values.","The Loeschian density constants $\\alpha_0^{(3)}$ and $\\beta_0$, which control the counting functions for Loeschian numbers and Loeschian sums of two squares, are now known to more than 100 digits.","The previously five-digit classical constants treated by the paper are now given to roughly 100 digits with explicit certified error bounds.","Products of the form $\\prod_{p\\in A} F(1/p)/G(1/p)$ over lattice-invariant classes, with $F(0)=G(0)=1$ and $F-G$ divisible by $X^2$, can be evaluated with a fully explicit error bound."],"supporting_citations":[{"why":"Supplies the $s \\to 2s$ iteration prototype for Euler products that the paper generalizes to arbitrary moduli and lattice-invariant classes.","marker":"[14]"},{"why":"Contains the classical product-over-characters identity (case $G_0=\\{1\\}$) that Lemma 2.2 extends to every subgroup.","marker":"[12]"},{"why":"Provides the incidence algebra and Möbius inversion formalism used to compute the inverse matrix $M_1^{-1}$.","marker":"[11]"},{"why":"Gives the original Witt decomposition that Lemma 6.1 turns into an infinite product expansion for $F(t)/G(t)$.","marker":"[18]"},{"why":"Offers a similar singular-series expansion that serves as the model for the rational-product version of the method.","marker":"[6]"}],"fun_headline_variants":["Euler product constants to 100+ digits fast","Multi-precision Euler products in double exponential time","Lattice-invariant Euler products computed to 100 digits","Fast certified Euler product constants for number theory","Double exponential Euler product computation hits 100 digits"],"cache_read_input_tokens":27392,"weakest_assumption_plain":"The speed guarantee depends on there being only a modest number of cyclic subgroups for each modulus; the authors could not prove that this number stays below a fixed power of the modulus.","fun_headline_variants_meta":{"raw":{"variants":["Euler product constants to 100+ digits fast","Multi-precision Euler products in double exponential time","Lattice-invariant Euler products computed to 100 digits","Fast certified Euler product constants for number theory","Double exponential Euler product computation hits 100 digits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000649,"raw_usage":{"total_tokens":2992,"prompt_tokens":973,"completion_tokens":2019,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":1946}},"tokens_in":589,"tokens_out":2019,"duration_ms":14302,"temperature":1.0,"reasoning_tokens":1946,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:34:35.102812+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $q$ equal to the product of many small primes, count the cyclic subgroups of $(\\mathbb{Z}/q\\mathbb{Z})^\\times$; a growth faster than any fixed power of $q$ would invalidate the claimed double-exponential speed.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the $s \\to 2s$ iteration prototype for Euler products that the paper generalizes to arbitrary moduli and lattice-invariant classes."},{"cited_title":"Le Math´ ematicien","cited_arxiv_id":null,"evidence_quote":"Contains the classical product-over-characters identity (case $G_0=\\{1\\}$) that Lemma 2.2 extends to every subgroup."},{"cited_title":"On the foundations of combinatorial t heory","cited_arxiv_id":null,"evidence_quote":"Provides the incidence algebra and Möbius inversion formalism used to compute the inverse matrix $M_1^{-1}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the original Witt decomposition that Lemma 6.1 turns into an infinite product expansion for $F(t)/G(t)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Offers a similar singular-series expansion that serves as the model for the rational-product version of the method."}],"review_version":1}