{"id":"9658602e-a981-4db1-b269-e0ddd8b4368a","arxiv_id":"2506.01865","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New series identities express quickly convergent sums with quadratic-irrational coefficients as pi squared or Dirichlet L-values L_d(2), including five cases beyond Guillera-Rogers.","lead":"This paper derives new, rapidly converging infinite series whose sums equal pi squared or values of Dirichlet L-functions for several negative discriminants. The identities extend an earlier framework to cases that were previously considered unsolvable.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The no-twist Kronecker application at non-fundamental discriminants in §3.2.3 is asserted without derivation; a direct high-precision check of (3.13) or (1.14) would settle the central claim.","rationale":"The reader's weakest_assumption already lists both Lemma 2.1 and the non-fundamental Kronecker application. I single out the latter as the more load-bearing issue because it is case-specific, asserted without derivation, and precisely the step that claims to go beyond Glasser–Zucker and Zucker–Robertson tables. Lemma 2.1, by contrast, is a single uniform reformulation that could be checked once against previously known Guillera–Rogers identities; the no-twist Kronecker step is a separate, unproved arithmetic identity. The paper contains no numerical verification of any kind, so the advertised results rest entirely on this chain. The identities are highly specific and cheap to test numerically, so the correct status is conditional rather than rejected: a single high-precision evaluation would either confirm or refute the central claim. I therefore keep the reader's CONDITIONAL verdict unchanged.","tokens_in":14806,"tokens_out":4461,"duration_ms":46393,"concrete_test":"Compute the left side of (3.13) for z = 1/2 + i√7 to, say, 50 significant digits by summing the defining Epstein zeta series (3.2) after unfolding into lattice sums, and compare with L_{−56}(2)L_8(2) computed independently from the Dirichlet series. If the match is not at the 1e-40 level, the no-twist Kronecker application fails and (1.14) is unsubstantiated. Alternatively, directly sum (1.14) with k up to ~60 and compare to the RHS; its ratio is about 0.084, so this is a cheap decisive check.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on exact evaluation of Re Σ_GR in §3. For the three cases with non-fundamental discriminants −448, −352, and −928 (Eqs. 3.8–3.10), the paper asserts that Kronecker's theorem applies \"without requiring additional twist factors.\" This is not a routine consequence of (3.7), which is stated only for fundamental discriminants (footnote 1, p. 11). Equations (3.13)–(3.14) then replace a difference of Epstein zeta values by products L_{d1}(2)L_{d2}(2) with no character/norm computation shown. If the implicit character identity is off by a twist, an L-value factor, or a sign, the corresponding Theorem 1.2 identities—in particular (1.14)—do not follow. Lemma 2.1 is also from an unpublished preprint, but it enters every case uniformly and can be cross-checked against the already accepted Guillera–Rogers framework; the no-twist Kronecker step is case-specific, unproved, and the exact place where the paper claims to go beyond existing tables. This is the most load-bearing unverified step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15041,"tokens_out":5449,"duration_ms":58231,"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":[{"comment":"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).","section":"§3.2.3, Eqs. (3.8)–(3.14)"},{"comment":"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.","section":"§2.1.1, Lemma 2.1"},{"comment":"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.","section":"§3.2.4, Eqs. (3.29)–(3.33)"}],"minor_comments":[{"comment":"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.","section":"Abstract and Theorem 1.2"},{"comment":"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.","section":"§3.2.2, Eq. (3.22)"},{"comment":"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.","section":"Tables 1–3 and numerical verification"}],"recommendation":"major_revision","confidential_remarks":"The paper's identities are likely correct, but the manuscript in its current form is not self-contained at exactly the points where it claims to go beyond the literature. The dependence on the unpublished preprint [11] and the unproved no-twist Kronecker application at non-fundamental discriminants should be the focus of the revision. I would ask the authors for full derivations of Lemma 2.1 and of equations (3.13)–(3.14), or, failing that, for a rigorous numerical verification with error bounds. The paper fits the journal's scope but needs these fixes before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives geometrically convergent series for L−56(2), L−68(2), L−87(2), L−111(2), and L−116(2), going beyond the solvable cases in Guillera–Rogers. The identities are explicit and checkable, and the paper is honestly organized: it says where each piece comes from, gives the CM points, and credits the earlier tables. That's real value, and the identities in Theorem 1.2 appear to be new.\n\nThe main soft spot is precisely the stress-test concern. In §3.2.3, for the three non-fundamental discriminants −448, −352, −928, the paper asserts that Kronecker's theorem applies “without requiring additional twist factors,” then states (3.13)–(3.14) with no character or norm computation shown. This is the load-bearing step that claims to go beyond existing tables. It may well be true, but as written it is an assertion, not a proof. A direct high-precision check of, say, (1.14) would settle it, and the paper includes no numerical verification at all. That is a gap, but it is fixable.\n\nTwo smaller issues: Lemma 2.1 comes from Zhou's unpublished preprint, and is not proved in the text. It is a reformulation of the Guillera–Rogers framework, so this is a provenance concern rather than a circular one. Also, several algebraic values are delegated to Zagier's algorithms without display, which is standard but makes independent verification slower.\n\nThe paper deserves a serious referee. The identities are concrete, the method is a known toolkit, and the gap is narrow enough that a referee can check it. I would not desk-reject. I'd ask for a derivation of (3.13)–(3.14) and at least one numerical spot-check before publication.","headline":"Concrete new series identities for five L-values, but the key Kronecker step for non-fundamental discriminants is asserted without proof.","tokens_in":15556,"tokens_out":2483,"would_cite":true,"duration_ms":26816,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11F03","11B65","05A19","33F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves fast-converging series identities for five Dirichlet L-values at 2, using Kronecker's theorem to split differences of lattice sums.","keywords":["Dirichlet L-functions","Ramanujan series upside-down","Kronecker's theorem","lattice sums","Epstein zeta functions","modular forms","geometric convergence","binomial coefficient series"],"falsifier":"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.","tokens_in":14606,"feed_emoji":"🧮","tokens_out":12998,"duration_ms":121889,"temperature":0.7,"pith_summary":"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.","feed_headline":"New exact series crack five stubborn number constants","feed_subtitle":"The identities turn L_{-56}(2) and four siblings into fast geometric binomial series, beyond known cases.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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)."],"supporting_citations":[{"why":"It supplies the upside-down series theory that the paper extends, including the solvable cases being surpassed and the dual identity (1.4).","marker":"[4]"},{"why":"It is quoted as Lemma 2.1 and provides the imaginary- and real-part conversion formulae for the transformed series.","marker":"[11]"},{"why":"It is the source of Kronecker's theorem used to rewrite lattice-sum combinations as products of Dirichlet L-values.","marker":"[5]"},{"why":"It provides tabulated lattice-sum values needed for the integer j-invariant cases.","marker":"[2]"},{"why":"It supplies additional lattice-sum evaluations and the companion Dirichlet theorem used alongside Kronecker's theorem.","marker":"[12]"},{"why":"It motivates the class of irrational series expressed through Dirichlet L-values at s=2.","marker":"[7]"},{"why":"It gives the algorithm for explicit algebraic values of the modular invariant and Legendre-Ramanujan function at CM points.","marker":"[8]"}],"fun_headline_variants":["Fast series compute five Dirichlet L-values beyond prior cases","Kronecker's theorem yields geometric series for five L(2,χ)","Exact series give L_{-56}(2) and four siblings","Guillera-Rogers plus Kronecker unlocks five L(2,χ) fast","Fast-converging identities for five Dirichlet L-values"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Fast series compute five Dirichlet L-values beyond prior cases","Kronecker's theorem yields geometric series for five L(2,χ)","Exact series give L_{-56}(2) and four siblings","Guillera-Rogers plus Kronecker unlocks five L(2,χ) fast","Fast-converging identities for five Dirichlet L-values"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001017,"raw_usage":{"total_tokens":4277,"prompt_tokens":915,"completion_tokens":3362,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":3267}},"tokens_in":531,"tokens_out":3362,"duration_ms":23460,"temperature":1.0,"reasoning_tokens":3267,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:31:49.312587+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Guillera and M","cited_arxiv_id":null,"evidence_quote":"It supplies the upside-down series theory that the paper extends, including the solvable cases being surpassed and the dual identity (1.4)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is quoted as Lemma 2.1 and provides the imaginary- and real-part conversion formulae for the transformed series."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the source of Kronecker's theorem used to rewrite lattice-sum combinations as products of Dirichlet L-values."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides tabulated lattice-sum values needed for the integer j-invariant cases."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies additional lattice-sum evaluations and the companion Dirichlet theorem used alongside Kronecker's theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It motivates the class of irrational series expressed through Dirichlet L-values at s=2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the algorithm for explicit algebraic values of the modular invariant and Legendre-Ramanujan function at CM points."}],"review_version":1}