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Fast converging irrational series for $ L(2,(\frac d\cdot))$

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves fast-converging series identities for five Dirichlet L-values at 2, using Kronecker's theorem to split differences of lattice sums.

desk verdict Concrete new series identities for five L-values, but the key Kronecker step for non-fundamental discriminants is asserted without proof. read the letter →

arxiv 2506.01865 v3 pith:4SJU7CPS submitted 2025-06-02 math.NT

Zhi-Wei Sun , Yajun Zhou This is my paper · ORCID
classification math.NT MSC 11M0611F0311B6505A1933F05
keywords DirichletL-functionsRamanujanseriesupside-downKronecker'stheoremlatticesumsEpsteinzetafunctionsmodularformsgeometricconvergencebinomialcoefficient
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 proves exact evaluations for a family of 'Ramanujan series upside-down': infinite series whose summands combine binomial coefficients, powers of algebraic numbers, and a linear factor $ak-b$. Some of these series are shown to equal rational multiples of $\pi^2$, and others to equal expressions built from Dirichlet $L$-values $L_d(2)=\sum_{k\ge1}\left(\frac d k\right)/k^2$. The new cases are the five values $L_{-56}(2)$, $L_{-68}(2)$, $L_{-87}(2)$, $L_{-111}(2)$, and $L_{-116}(2)$, which were not among the previously solvable lattice-sum cases. Because each series has a geometric ratio strictly inside the unit circle, the identities give rapidly converging numerical recipes for those constants.

What carries the argument

The load-bearing object is the transformed series $\Sigma^{\mathrm{GR}}_{\nu}(z)$ for $\nu=-1/4,-1/3,-1/2$, together with the modular invariant $\alpha_N(z)$ built from the Dedekind eta function and the Legendre-Ramanujan function $R_\nu$ (an auxiliary function encoding the linear coefficient of the series). Lemma 2.1 supplies the two conversion formulae used throughout: the imaginary part of $\Sigma^{\mathrm{GR}}_{\nu}(z)$ is a rational multiple of $\pi^2$, and the real part equals $\frac{8\pi^2}{3}$ times a difference of Epstein zeta functions. Kronecker's theorem then converts suitable linear combinations of those zeta differences into products of Dirichlet $L$-values, and the paper chooses CM points so that the combinations reduce to the five target values.

What would settle it

Evaluate both sides of equation (1.14) numerically: compute the left-hand series to high precision using its geometric ratio near $0.104$, and compute $L_{-56}(2)$ independently by summing $\sum_k (\frac{-56}{k})/k^2$ directly; any disagreement beyond rounding would refute the claimed identity. Repeating the same check on (1.13), (1.22), and (1.25) against independently summed $L$-values would settle the theorem as a whole.

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Extended reading notes

Core claim

The paper's central claim is that the upside-down series reformulation, combined with Kronecker's theorem, evaluates the real part of the transformed series $\Sigma^{\mathrm{GR}}_{\nu}(z)$ even when the two lattice sums entering the formula are not individually known closed forms. The key step is to express the difference $E_{\Gamma_0(1)}(Nz,2)-E_{\Gamma_0(1)}(z,2)$ through combinations of Dirichlet $L$-values, using complex-multiplication (CM) points whose $j$-invariants share a minimal polynomial. Working out this arithmetic for $N=4$, $3$, and $2$ yields fourteen explicit identities, such as $$\sum_{k=1}^\infty \frac{6(17\sqrt7+35)k-35\sqrt7-89}{$k^{3}$\binom{2k}{k}^3}(-211)^k(45-17\sqrt7)^{2k}=128\left(20L_{-8}(2)-7\sqrt7\,L_{-56}(2)\right).$$ Rearranging with known constants isolates the five target $L$-values, going beyond the individually solvable lattice-sum cases catalogued in earlier tables.

Load-bearing premise

The load-bearing premise is that a conversion formula quoted from an unpublished preprint is correct, and that a classical identity relating sums of lattice functions may be applied to certain discriminants with square factors without inserting extra correction factors; if either use is invalid, the new identities fail even though the series themselves are well-defined.

Editorial extensions

If this is right

  • Each displayed identity yields a fast numerical route to the corresponding target $L$-value: substituting known constants such as $G$ and $L_{-8}(2)$ and rearranging algebraically isolates $L_{-56}(2)$, $L_{-68}(2)$, $L_{-87}(2)$, $L_{-111}(2)$, or $L_{-116}(2)$.
  • The identities provide independent numerical checks on any computation of Dirichlet $L$-values for these discriminants.
  • The method widens the set of lattice-sum differences expressible through $L$-values beyond the individually solvable cases catalogued in earlier tables.
  • Theorem 1.1 and its remarks give new closed forms for series involving Fibonacci and Lucas numbers, such as equations (1.6) and (1.7).

Reading between the lines

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

  • Beyond the paper: the same difference-of-lattice-sums splitting is likely to work for other non-fundamental discriminants whose ideal class groups are small, provided the relevant twist factors are computed explicitly rather than assumed absent.
  • Beyond the paper: since the right-hand sides are finite $\mathbb{Q}$-linear combinations of $L$-values, these identities could serve as test inputs for conjectural relations among Dirichlet $L$-values at $s=2$.
  • Beyond the paper: a systematic search over CM points of the form used here, checking numerically whether the Kronecker product formula holds without twist factors, could produce additional fast series for other $L_d(2)$ constants.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies Ramanujan upside-down series whose summands contain quadratic irrationals. Theorem 1.1 gives two series evaluating to rational multiples of π^2, and Theorem 1.2 gives ten identities expressing such series as explicit linear combinations of Dirichlet L-values L_d(2), including combinations that involve L_{-56}(2), L_{-68}(2), L_{-87}(2), L_{-111}(2), and L_{-116}(2). The method combines a reformulation of the Guillera–Rogers theory (Lemma 2.1, quoted from the second author's preprint [11]) with reductions of differences of Epstein zeta values to products of Dirichlet L-values via Kronecker's theorem.

Significance. If fully established, the paper would provide a useful extension of the Guillera–Rogers tables, giving explicit geometrically convergent series with algebraic parameters for combinations of L-values that were not previously treated as 'solvable' lattice sums. The identities are concrete and numerically checkable, and the tables of algebraic parameters are valuable. However, the two main analytical pillars of the proof are not self-contained: Lemma 2.1 is imported from an unpublished preprint, and the application of Kronecker's theorem to non-fundamental discriminants in §3.2.3 is asserted without derivation. These gaps affect the central claim of going beyond the known solvable cases, so the paper's correctness is not yet established in the form submitted.

major comments (3)
  1. [§3.2.3, Eqs. (3.8)–(3.14)] The claim that Kronecker's theorem applies to the non-fundamental discriminants -448, -352, and -928 'without requiring additional twist factors' is the pivotal step for the first three identities in Theorem 1.2(a), but it is not proved. Equation (3.7), the stated form of Kronecker's theorem, is restricted to fundamental discriminants in footnote 1, and the discriminants in (3.8)–(3.10) are not fundamental. The paper does not exhibit the character/norm factors that would arise in the non-fundamental case, nor does it provide a numerical check of (3.13) and (3.14). Since these equations are used to derive (1.12)–(1.14), this gap is load-bearing for the paper's main claim. The authors should supply the missing derivation or include a certified high-precision verification of at least one representative identity such as (1.14).
  2. [§2.1.1, Lemma 2.1] The central conversion formula (2.8), which expresses Im Σ_GR^ν(z) in terms of Eisenstein series, is quoted from the unpublished preprint [11, Theorem 4.3] without proof. This lemma is used in every evaluation in the paper, including both Theorem 1.1 and Theorem 1.2. The manuscript should be made self-contained: either prove Lemma 2.1 directly or state the exact hypotheses and provide a publicly accessible reference with a complete proof. As written, the correctness of all main results depends on an unreviewed preprint by the second author.
  3. [§3.2.4, Eqs. (3.29)–(3.33)] The two 'trickier cases' in Table 2 (discriminants -192 and -96) rely on additional twist factors 9/8 and 11/8 in equations (3.32) and (3.33), quoted from [2,12]. Since these factors are essential for identities (1.16) and (1.17), the paper should show the character computation or at least verify these constants, rather than citing tables whose non-fundamental entries involve the same kind of unproved step the paper claims to handle without twists.
minor comments (3)
  1. [Abstract and Theorem 1.2] The abstract states that the paper obtains 'geometrically convergent series for L_{-56}(2)' and similar values, but Theorem 1.2 actually gives series for linear combinations such as 20L_{-8}(2) − 7√7 L_{-56}(2) in (1.14). The authors should rephrase the abstract or add a short linear-algebra argument showing how a single L-value can be isolated.
  2. [§3.2.2, Eq. (3.22)] Equation (3.22) is called a companion to (3.7), but no context is given for the distinction between fundamental and non-fundamental discriminants; in particular, the meaning of ζ(2)L_D(2) and the excluded cases D = −3, −4 should be stated explicitly.
  3. [Tables 1–3 and numerical verification] The tables are dense and would benefit from an explicit note defining the notation L_{d1}(2)L_{d2}(2) and from a statement indicating which columns are exact equalities versus evaluations. A short appendix with high-precision decimal checks of all ten identities in Theorem 1.2 would substantially increase confidence, given the reliance on unpublished or tabular sources.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: L-values appear as Kronecker-outputs, not fitted inputs; the same-author reformulation is independent support.

full rationale

The claimed derivation is not circular. The summand constants in Theorem 1.2 are fixed by the CM point choice and computed from alpha_N and R_nu via the stated algorithms, not fitted to the target L-values; those L-values occur only after Kronecker's theorem is applied to lattice-sum differences (3.13)-(3.14). Lemma 2.1/3.2 is quoted from the second author's preprint [11, Thm 4.3], but it is a parameter-free reformulation of the externally published Guillera-Rogers theory [4, (46)] with stated hypotheses (2.7) that do not include the Theorem 1.2 identities, so under the rubric it is independent support and does not make the argument circular. The genuinely unverified step is the Section 3.2.3 assertion that Kronecker's theorem works without twist factors for the non-fundamental discriminants -448, -352, -928; that is a correctness risk, not a circular reduction of the claim to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. The constants in the series are algebraic numbers determined by CM points and modular forms, and the L-values on the right-hand sides are targets, not fitting inputs. The main assumptions are the correctness of the coauthor's prior reformulation, the cited lattice-sum tables, and the application of Kronecker's theorem to non-fundamental discriminants.

assumptions (5)
  • domain assumption Lemma 2.1: for z satisfying (2.7), the imaginary part of the Guillera-Rogers series is given by the explicit formula (2.8), and the real part by (3.3).
    The paper's proofs of Theorems 1.1 and 1.2 depend on this evaluation formula, which is quoted from [11, Theorem 4.3], an unpublished preprint by the second author, and is not proved in this paper.
  • standard math Kronecker's theorem in the form (3.7) and Dirichlet's companion theorem (3.22) connect sums of Epstein zeta values at CM points to products of Dirichlet L-values.
    Quoted from Siegel [5, Theorem 4] and Zucker-Robertson [12, (2.3)], and used throughout Section 3 to evaluate differences of lattice sums.
  • domain assumption The tabulated lattice-sum values from Glasser-Zucker [2] and Zucker-Robertson [12] are correct.
    These tables supply E_Gamma0(1)(w,2) at specific integer or quadratic CM points, e.g., (3.16), (3.18), (3.34), and (3.50), with no independent verification in the paper.
  • domain assumption The algorithms in Zagier [8, Section 6] compute the algebraic numbers alpha_N(z) and R_nu(1-2 alpha_N(z)) exactly.
    Special values such as (2.14)-(2.15) and the entries in Tables 1-3 rely on these algorithms; the results are asserted without showing the computations.
  • domain assumption The Guillera-Rogers theory, as developed in [4], including Proposition 3 and the lattice-sum identity (4.46), correctly describes the real parts of the upside-down Ramanujan series.
    This is the starting point for all evaluations and is external to the present paper, but it is invoked without restating or proving the underlying theory.

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Pith. "Pith review of Fast converging irrational series for $ L(2,(\frac d\cdot))$." pith.science (2026). https://pith.science/paper/4SJU7CPS

@misc{pith2026250601865,
  author       = {Pith},
  title        = {Pith review of: Fast converging irrational series for $ L(2,(\frac d\cdot))$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4SJU7CPS}},
  note         = {Machine review of arXiv:2506.01865}
}
abstract

By exploring the theory of Guillera-Rogers, we evaluate some infinite series whose summands are quadratic irrationals, in terms of $\pi$ and special values of Dirichlet $L$-functions $ L_d(2)\equiv L(2,(\frac d\cdot)):=\sum_{k=1}^\infty\left( \frac{d}{k} \right)\frac1{k^2}$. Applying Kronecker's theorem to linear combinations of lattice sums, we obtain geometrically convergent series for $ L_{-56}(2)$, $ L_{-68}(2)$, $ L_{-87}(2)$, $ L_{-111}(2)$, and $ L_{-116}(2)$, which go beyond the solvable cases of Guillera-Rogers.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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