Fast multi-precision computation of some Euler products

L\'eon Surel; Olivier Ramar\'e; Salma Ettahri

REVIEW 3 major objections 5 minor 18 references

Fast multi-precision computation of some Euler products

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The authors prove Euler products over lattice-invariant classes can be computed in double-exponential time, with explicit error bounds.

desk verdict 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. read the letter →

arxiv 1908.06808 v2 pith:QJOG4K6C submitted 2019-08-19 math.NT math.CO

Salma Ettahri , Olivier Ramaré , Léon Surel This is my paper · ORCID

classification math.NTmath.CO MSC 11Y6011N1305A

keywords Eulerproductslattice-invariantclassesLoeschiannumbersLal'sconstantmulti-precisioncomputationMoebiusinversionWittdecompositiondouble-exponentialtime

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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 $q$.

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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

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.

What'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.

Other 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.

Who'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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Theorem 1.3, Eq. (12); Section 1, Theorem 1.2] 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 4, Lemma 4.2] 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 7, Proposition 7.3 and following paragraph] 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.

minor comments (5)

[Theorem 1.3 statement] 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.
[Equation (6) and surrounding text] 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.
[Theorem 1.4, bound (39)] 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 9, Modulo 13 row] 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.
[Abstract and Section 1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the target Euler products are obtained from an explicit recursion whose inputs are Dirichlet L-values/Hurwitz zeta data, not from the quantities being predicted.

full rationale

The derivation is self-contained and non-circular. The target vector V_s(1) is not assumed: it is obtained by solving the exact identity Gamma(t) = sum_d M_d V_s(dt) (Eq. 21) for V_s(t) via the explicitly inverted matrix M_1^{-1} (Proposition 4.1), giving a recursion whose input at each stage is Gamma_s(d1...dv t), computed from Dirichlet L-functions/Hurwitz zeta values, an external source independent of the Euler products being predicted. The residual V_s(D) terms are controlled by Lemma 4.3, an independent tail estimate, not by fitting the answer. Theorem 1.4 likewise reduces F(1/p)/G(1/p) to zeta_P(j;q,A|r) via the Witt/Newton-Girard decomposition, and the error bounds are explicit. The numerical checks against direct truncation of the Euler product and against published constants are verification, not fitted inputs. The paper even flags its own open issue: 'we have ... not been able to establish that |G7| <<_epsilon q^epsilon' (Section 7), which is a complexity caveat, not a circularity. Any concern that the stated error bound does not prove double-exponential convergence for all q is a correctness/complexity risk, not a reduction of the predicted quantity to its own definition.

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

No free parameters are fitted; the inputs are q, F, G, and accuracy parameters P, r, J. The axioms are standard background (character orthogonality, Hurwitz zeta, Witt decomposition, Möbius inversion). The main domain assumption is that A is a lattice-invariant class, which is the scope of the paper.

assumptions (4)

standard math Characters of the finite abelian group (Z/qZ)^× separate points and satisfy orthogonality, used to derive the factorization identities in Lemma 2.2.
This is textbook group theory (Serre, ref [12]); the paper invokes it in the proof of Lemma 2.2.
domain assumption Values of the Hurwitz zeta function can be computed to arbitrary precision with rigorous error control, used by the function GetGamma to evaluate L_P(s,χ).
The paper states Γ_s is computed 'from the values of the Hurwitz zeta function' (Section 1, implementation notes) but does not analyze or cite a specific high-precision Hurwitz-zeta algorithm.
standard math The Witt decomposition (35) holds for real polynomials with F(0)=1, relying on Newton-Girard identities and Lambert series manipulation.
Proved sketchily in Lemma 6.1 following [6, Lemma 1]; the ingredients are standard.
standard math The Möbius function of the subgroup lattice of a cyclic group is μ(|H/K|), reducing the inversion to integer Möbius inversion.
Proved in equation (27) via chains; standard Rota incidence algebra.

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Pith. "Pith review of Fast multi-precision computation of some Euler products." pith.science (2026). https://pith.science/paper/QJOG4K6C

@misc{pith2026190806808,
  author       = {Pith},
  title        = {Pith review of: Fast multi-precision computation of some Euler products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QJOG4K6C}},
  note         = {Machine review of arXiv:1908.06808}
}

abstract

For every modulus $q\ge3$, we define a family of subsets $\mathcal{A}$ of the multiplicative group $(\mathbb{Z}/{q}\mathbb{Z})^\times$ for which the Euler product $\prod_{p\text{mod}q\in\mathcal{A}}(1-p^{-s})$ can be computed in double exponential time, where $s>1$ is some given real number. We provide a Sage script to do so, and extend our result to compute Euler products $\prod_{p\in\mathcal{A}}F(1/p)/G(1/p)$ where $F$ and $G$ are polynomials with real coefficients, when this product converges absolutely. This enables us to give precise values of several Euler products intervening in Number Theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [1]

Steven R. Finch. Mathematical constants, volume 94 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2003

work page 2003

[2] [2]

R epresentation of integers by cyclotomic binary forms

´Etienne Fouvry, Claude Levesque, and Michel W aldschmidt. R epresentation of integers by cyclotomic binary forms. Acta Arith., 184(1):67–86, 2018

work page 2018

[3] [3]

Gourdon and P

X. Gourdon and P. Sebah. Constants from number theory. http: // numbers. computation. free. fr/ Constants/ constants. html , 2010. http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.ps

work page 2010

[4] [4]

A classical introduction to modern number theory , vol- ume 84 of Graduate Texts in Mathematics

Kenneth Ireland and Michael Rosen. A classical introduction to modern number theory , vol- ume 84 of Graduate Texts in Mathematics . Springer-Verlag, New York, second edition, 1990

work page 1990

[5] [5]

M. Lal. Primes of the form n4 ` 1. Math. Comp. , 21:245–247, 1967

work page 1967

[6] [6]

P. Moree. Approximation of singular series constant and automata. with an appendix by gerhard niklasch. Manuscripta Matematica , 101(3):385–399, 2000

work page 2000

[7] [7]

Moree and R

P. Moree and R. Osburn. Two-dimensional lattices with fe w distances. Enseign. Math. (2) , 52(3-4):361–380

work page

[8] [8]

On the average number of elements in a ﬁnite ﬁeld with order or index in a prescribed residue class

Pieter Moree. On the average number of elements in a ﬁnite ﬁeld with order or index in a prescribed residue class. Finite Fields Appl. , 10(3):438–463, 2004

work page 2004