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Multiple Dirichlet series predictions for moments of $L$-functions: unitary, symplectic and orthogonal examples

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Pith's one-line read Multiple Dirichlet series predicts L-function moments and fixes an elliptic-curve failure

desk verdict The claimed elliptic-curve failure of CFKRS is a sign error away from disappearing; the rest of the paper is a solid MDS heuristic framework. read the letter →

arxiv 2501.12529 v1 pith:MBHFIEN7 submitted 2025-01-21 math.NT

Siegfred Baluyot , Martin Čech This is my paper · ORCID
classification math.NT MSC 11M06
keywords multipleDirichletseriesmomentsofL-functionsCFKRSrecipequadratictwistsellipticcurvesrootnumbersymplecticfamilyunitaryshifted
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

Using multiple Dirichlet series, the paper predicts asymptotic formulas for shifted moments of four families of L-functions: even primitive Dirichlet characters (unitary), quadratic Dirichlet L-functions (symplectic), quadratic twists of a fixed modular form (orthogonal), and quadratic twists of a fixed elliptic curve. For the first three families, the prediction agrees term by term with the CFKRS recipe. For the elliptic-curve family, the recipe as usually stated is wrong, because averaging root numbers separately from the Dirichlet coefficients drops a correlation; the paper gives a corrected version and verifies the correction on the first moment. The main structural finding is a one-to-one correspondence between the residues of the multiple Dirichlet series and the terms of the recipe prediction. If the paper's continuation conjectures hold, each family's moment formula follows with a power-saving error term.

What carries the argument

The central object is the multiple Dirichlet series built from moments of the family's L-functions, for example $A_{M,N}(s_1,\ldots,s_k;z_1,\ldots,z_k;w)=\sum_{q\ge 1}\sum_{\chi\bmod q}^{\flat}\chi(M)\chi(N)L(s_1,\chi)\cdots L(z_k,\chi)/q^w$, with analogous sums for quadratic characters and quadratic twists. Its pole at $w=2$ or $w=1$ has residue equal to the diagonal main term of the recipe, and the functional equations of the L-functions shift that pole to locations $\sigma_{J,H}(w)=2$ or $\sigma_J(w)=1$ whose residues match the remaining recipe terms. The load-bearing conjecture is that the series continues meromorphically to a tube domain (a multidimensional analogue of a vertical strip) containing the central point, with only those shifted poles and with polynomial growth; the paper verifies convergence and the pole-residue statements only in smaller regions.

What would settle it

Evaluate the weighted first moment of $L(1/2+\alpha,E_d)$ for a fixed elliptic curve of non-square conductor. The unmodified recipe predicts only the diagonal term, while the corrected prediction adds the swap term $X^{2-2s} X_E(s)\tilde g(2-2s)H_{E,MN}(1-s)$ with $s=1/2+\alpha$; an asymptotic or high-precision numerical evaluation would reveal which term is present, and the paper's appendix says the extra term appears.

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Extended reading notes

Core claim

The central claim is that the multiple Dirichlet series heuristic and the CFKRS recipe produce the same shifted-moment asymptotics, term by term, for the unitary, symplectic and orthogonal families, and that the elliptic-curve family is the one place where an unmodified recipe fails. The mechanism is a pole at $w=2$ (unitary) or $w=1$ (the other families) whose residue is the diagonal term, plus additional poles created by applying the functional equations of the individual L-functions; the residues of those shifted poles are exactly the swap terms of the recipe. In the elliptic-curve case, the functional equations introduce a factor $\chi_d(N)^{|J|}$, so the shifted series is the same multiple Dirichlet series with $M$ replaced by $M N^{|J|}$; the usual recipe step of averaging root numbers first is what loses this factor. The paper states the precise continuation assumptions (Conjectures 2.6, 3.3, 4.3 and 5.5) under which these residue computations become asymptotics, and the appendix shows for the first moment that the corrected prediction holds while the unmodified one does not.

Load-bearing premise

The predictions hinge on the unproved assumption that every multiple Dirichlet series under study extends meromorphically into a large enough complex region around the critical point, with only the expected shifted poles and polynomial growth; the rigorous results in the paper reach that region only partially.

Editorial extensions

If this is right

  • Conditional on the continuation conjectures, Propositions 2.8, 3.4, 4.4 and 5.6 turn the residue computation into asymptotic formulas with power-saving error terms, matching the CFKRS predictions for the first three families.
  • For the elliptic-curve family, the unmodified Conjecture 5.1 fails at the first moment, while the modified Conjecture 5.2 reproduces the asymptotic formula obtained in the appendix.
  • Every admissible subset of L-functions flipped by the approximate functional equation corresponds to one pole of the multiple Dirichlet series, so the recipe's swap terms have a uniform analytic origin.
  • The same machinery is remarked to extend to other families and to related statistics such as Ratios conjectures and one-level density of low-lying zeros.

Reading between the lines

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

  • If the continuation conjectures hold beyond the leading pole, the multiple Dirichlet series should also encode secondary lower-order terms of the type already found for the third moment of quadratic Dirichlet L-functions, terms that the recipe does not predict.
  • The root-number and coefficient correlation exposed here is likely present in every family with genuinely varying root numbers, such as quadratic twists of higher-level newforms or twists restricted to arithmetic progressions, where the corrected averaging prescription would also be needed.
  • The residue-to-term dictionary suggests a practical route for generating new moment predictions: compute shifted poles of the multiple Dirichlet series rather than executing the full recipe.
  • A second-moment computation for elliptic-curve twists would sharply discriminate between the two versions of the recipe, since at that order the two predictions differ by several terms, not just one.
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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 / 5 minor

Summary. The paper develops multiple Dirichlet series (MDS) heuristics for shifted moments of four families of L-functions: even primitive Dirichlet characters of conductor up to Q, quadratic Dirichlet L-functions, quadratic twists of a level-one Hecke eigencuspform, and quadratic twists of a fixed elliptic curve with non-square conductor. For the first three families, the authors compute the residues of the associated multiple Dirichlet series and show a term-by-term match with the CFKRS recipe prediction. For the elliptic-curve family, they claim that the unmodified CFKRS recipe misses terms with an odd number of functional-equation applications, because the root number ε(E)χ_d(−N) is correlated with the Dirichlet coefficients; they propose a modified recipe and state that the appendix verifies the first moment by adapting a result of Shen. The asymptotic predictions are explicitly conditional on strong meromorphic-continuation conjectures, and the rigorous content is mainly the residue computations in Propositions 2.4, 3.2, 4.2, and 5.4.

Significance. If the claims are correct, the paper provides a useful dictionary between residues of multiple Dirichlet series and terms of the CFKRS recipe, and it identifies a genuine subtlety in the recipe for families with sign-coefficient correlation. The residue computations are standard and appear correct, and the conjectured analytic properties are stated precisely enough to be falsifiable. The main weakness is that the appendix's verification of the elliptic-curve first moment is only sketched; since the paper's central novelty is the failure of the unmodified recipe, this verification is load-bearing and needs to be either fully supplied or explicitly downgraded to a conjecture.

major comments (3)
  1. [§6.2, Eq. (6.2.6)] The appendix claims that a 'straightforward modification' of Shen's proof yields the asymptotic formula (6.2.6) with a power-saving error term, but the required modifications are not actually given. The function R_E(α, ℓ) is never defined, the treatment of bad Euler factors at primes dividing N is not explained, and the passage from Shen's χ_{8d}-family to the family of all positive fundamental discriminants coprime to N is not justified. Since the assertion that Conjecture 5.1 is false for the first moment rests entirely on (6.2.6), this gap is load-bearing and must be addressed before the paper's central claim can be accepted.
  2. [§5.1–5.2, (5.1.4), (5.2.10)] Equations (5.1.4), (5.2.10), and the appendix use χ_d(N) where the functional equation (5.1) has χ_d(−N). The equality χ_d(−N)=χ_d(N) does hold because d runs over positive fundamental discriminants, for which χ_d(−1)=1, but this fact is never stated. The odd-|J| terms in Conjecture 5.2 depend on this point, and the current notation invites a spurious sign-error objection; please add an explicit remark and use a single convention throughout.
  3. [§5.1, Conjecture 5.1] The paper states that Conjecture 5.1 is false by comparison with (6.2.6). However, (6.2.6) is itself derived only under an unproved extension of Shen's theorem and under the meromorphic-continuation Conjecture 5.5. The falsity claim should be framed as conditional on these two unproved inputs unless the appendix is made fully rigorous.
minor comments (5)
  1. [Eq. (5.1.4)] The displayed formula uses a product over J, but it should be a sum over J; the surrounding text and the subsequent equations clearly intend a sum.
  2. [Conjectures 2.6 and 4.3] The subset notation ⊂ should be ⊆; otherwise the poles corresponding to applying the functional equation to all L-factors are not removed by the product.
  3. [Appendix, after (6.2)] The approximate functional equation is written for L(1/2+α, E_d), but in the definition of M_E(α, ℓ) the argument is E_{8d}; please correct this to E_{8d} throughout the appendix.
  4. [References] Reference [31] contains a formatting error in the authors' name ('Radziwi/suppress l/suppress l'); this should be fixed.
  5. [Appendix] The functions R_E(α, ℓ) and R_E(−α, ℓN) are used without definition; please define them explicitly and relate them to H_{E,M} from (5.1.6).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the multiple Dirichlet series residues are computed from the L-function functional equations, and the comparison to the CFKRS recipe is made afterwards.

full rationale

The derivation chain is self-contained. Each family begins with an independently defined multiple Dirichlet series, proves a residue formula in a region of absolute convergence from orthogonality and known Dirichlet series facts, and then applies the functional equation of the underlying L-functions to locate shifted poles; the resulting residues are algebraically equal to the CFKRS recipe terms, but the recipe is not used as an input in their computation. The central continuation conjectures are explicitly labeled conjectures rather than consequences of CFKRS. Technical input from the same authors' earlier work, such as [7, Lemma 2.4] for the Dirichlet series L_D(w, chi), is a published theorem with independent proof and does not assume the target asymptotics. The elliptic-curve modification is transparently described as suggested to match the multiple Dirichlet series heuristic and is checked against Shen's theorem in the Appendix, an external benchmark. No fitted parameter is renamed as a prediction. A separate sign inconsistency in (5.2.10) is a correctness concern, not a circularity, and does not change this verdict.

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

The paper introduces no fitted constants and no new postulated objects. Its central predictions rest on unproved meromorphic continuation conjectures and on heuristic root-number averaging assumptions, both explicitly stated. The appendix's first moment verification is an outlined extension of Shen's proof rather than a complete derivation.

assumptions (5)
  • ad hoc to paper The multiple Dirichlet series A_{M,N}, A_M, A_{f,M} and A_{E,M} have holomorphic continuation to tube domains containing the central point with only the shifted poles and polynomial boundedness.
    This is the load-bearing heuristic assumption stated as Conjectures 2.6, 3.3, 4.3 and 5.5. Without it the contour shift giving the asymptotic formulas cannot be justified; known meromorphic continuation is limited to restricted regions such as Re(w)>1 and Re(s_j)>5/4.
  • domain assumption Root-number averages vanish unless the number of swapped factors satisfies |J|=|H| for the unitary family and 2 divides |J| for the orthogonal families; for elliptic curve twists the product of root number and coefficients must be averaged jointly rather than separately.
    This discards the off-diagonal and oscillating terms. For the elliptic curve family the separate averaging is false, and the joint averaging is the correction; the paper proves the failure for k=1 via the appendix.
  • domain assumption The approximate functional equations (2.3), (3.2), (4.3) and (5.2) can be used with sums extended to infinity after discarding off-diagonal terms.
    This is the CFKRS recipe step. The paper acknowledges that the individual steps cannot be justified, citing Note (6) on page 21 of Conrey, Farmer, Keating, Rubinstein and Snaith.
  • standard math Known analytic estimates are valid, including orthogonality of Dirichlet characters, bounds for L_D(w, chi) from Cech's work, Deligne's bound for Hecke eigenvalues, the Hasse bound, the modularity theorem and the convexity bound.
    Used in Propositions 2.4, 3.2, 4.2 and 5.4 to establish absolute convergence and meromorphic continuation in limited regions; these are standard results or cited from the literature.
  • ad hoc to paper Shen's proof of the first moment for level-one modular forms extends to general conductor N with a straightforward modification.
    Used in the appendix to derive formula (6.2.6) for elliptic curve twists. The modification is only sketched, not fully carried out.

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Pith. "Pith review of Multiple Dirichlet series predictions for moments of $L$-functions: unitary, symplectic and orthogonal examples." pith.science (2026). https://pith.science/paper/MBHFIEN7

@misc{pith2026250112529,
  author       = {Pith},
  title        = {Pith review of: Multiple Dirichlet series predictions for moments of $L$-functions: unitary, symplectic and orthogonal examples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MBHFIEN7}},
  note         = {Machine review of arXiv:2501.12529}
}
abstract

We devise heuristics using multiple Dirichlet series to predict asymptotic formulas for shifted moments of (1) the family of Dirichlet $L$-functions of all even primitive characters of conductor $\leq Q$, with $Q$ a parameter tending to infinity, (2) the family of quadratic Dirichlet $L$-functions, (3) the family of quadratic twists of an $L$-function associated to a fixed Hecke eigencuspform for the full modular group, and (4) the family of quadratic twists of an $L$-function of a fixed arbitrary elliptic curve over $\mathbb{Q}$ that has a non-square conductor. For each of these families, the resulting predictions agree with the predictions of the recipe developed by Conrey, Farmer, Keating, Rubinstein, and Snaith, except for (4), where the recipe requires a slight modification due to a correlation between the Dirichlet coefficients and the root number of the corresponding $L$-functions. We find a one-to-one correspondence between the residues from the multiple Dirichlet series analysis and the terms from the recipe prediction.

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Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Baluyot and C

    S. Baluyot and C. L. Turnage-Butterbaugh. A mean value theorem for Dirichlet polynomials associated with primitive Dirichlet L-functions. Int. Math. Res. Not. to appear

    work page

  2. [2]

    Breuil, B

    C. Breuil, B. Conrad, F. Diamond, and R. Taylor. On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc., 14 (4) (2001), 843–939

    work page 2001

  3. [3]

    Bucur and A

    A. Bucur and A. Diaconu. Moments of quadratic Dirichlet L-functions over rational function fields . Mosc. Math. J., 10 (3) (2010), 485–517, 661

    work page 2010

  4. [4]

    D. Bump. Introduction: multiple Dirichlet series . In: Multiple Dirichlet series, L-functions and auto- morphic forms, Progr. Math., vol. 300, Birkh¨ auser/Springer, New York, 2012, pp. 1–36

    work page 2012

  5. [5]

    D. Bump, S. Friedberg, and J. Hoffstein. On some applications of automorphic forms to number theory , Bull. Amer. Math. Soc. 33 (1996), 157–175

    work page 1996

  6. [6]

    M. ˇCech. Applications of multiple Dirichlet series in analytic numb er theory. Ph.D. Thesis, Concordia University, Montreal, Quebec, 2022

    work page 2022

  7. [7]

    M. ˇCech. The ratios conjecture for real Dirichlet characters and mul tiple Dirichlet series . Trans. Amer. Math. Soc. 377 (2024), no. 5, 3487–3528

    work page 2024

  8. [8]

    M. ˇCech. Moments of real Dirichlet L-functions and multiple Dirichlet series . Preprint, available at https://arxiv.org/abs/2402.07473

    work page Pith review arXiv

Show all 41 references
  1. [9]

    J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith. Integral moments of L-functions. Proc. London Math. Soc. (3) 91 (2005), 33–104

    2005

  2. [10]

    Conrey, H

    B. Conrey, H. Iwaniec, and K. Soundararajan. The mean square of the product of a Dirichlet L-function and a Dirichlet polynomial . Funct. Approx. Comment. Math. 61 (2) (2019), 147–177

    2019

  3. [11]

    J. B. Conrey, B. Rodgers. Averages of quadratic twists of long Dirichlet polynomials . Preprint, available at https://arxiv.org/abs/2208.01783

    arXiv

  4. [12]

    J. B. Conrey, N. Snaith, Applications of the L-functions Ratios conjecture , Proc. Lond. Math. Soc. 93 (3) (2007), 594–646

    2007

  5. [13]

    A. Diaconu. On the third moment of L( 1 2 , χ d) I: The rational function field case . J. Number Theory 198 (2019), 1–42

    2019

  6. [14]

    Diaconu, P

    A. Diaconu, P. Garrett, and D. Goldfeld. Natural boundaries and integral moments of L-functions. In: Multiple Dirichlet series, L-functions and automorphic forms, Progr. Math., vol. 300, Birkh¨ auser/Springer, New York, 2012, pp. 147–172

    2012

  7. [15]

    Diaconu and D

    A. Diaconu and D. Goldfeld. Second moments of quadratic Hecke L-series and multiple Dirichlet series I. In: Multiple Dirichlet Series, Automorphic Forms, and Analytic Numbe r Theory (Bretton Woods, New Hampshire, 2005), Proc. Sympos. Pure Math. 75, Amer. Math. Soc., Providence...

    2005

  8. [16]

    Diaconu, D

    A. Diaconu, D. Goldfeld, and J. Hoffstein. Multiple Dirichlet series and moments of zeta and L-functions. Compos. Math. 139 (2003), 297–360

    2003

  9. [17]

    Diaconu and H

    A. Diaconu and H. Twiss. Secondary terms in the asymptotics of moments of L-functions. J. Number Theory 252 (2023), 243–297

    2023

  10. [18]

    Diaconu and I

    A. Diaconu and I. Whitehead. On the third moment of L( 1 2 , χ d) II: The number field case . J. Eur. Math. Soc. 23 (2021), 2051–2070

    2021

  11. [19]

    Fisher and S

    B. Fisher and S. Friedberg. Double Dirichlet series Over function fields . Compos. Math. 140 (3) (2004), 613–630

    2004

  12. [20]

    A. M. Florea. The second and third moment of L(1/ 2, chi ) in the hyperelliptic ensemble . Forum Math. 29 (4) (2017) 873–892

    2017

  13. [21]

    Friedberg and J

    S. Friedberg and J. Hoffstein. Nonvanishing theorems for automorphic L-functions on GL(2) . Ann. of Math. 142 (1995), 385–423

    1995

  14. [22]

    Goldfeld and J

    D. Goldfeld and J. Hoffstein. Eisenstein series of 1/2-integral weight and the mean value of real Dirichlet L-series. Invent. Math. 80 (1985), 185–208

    1985

  15. [23]

    A. Good. The convolution method for Dirichlet series . In: The Selberg Trace Formula and Related Topics (Brunswick, Maine, 1984), Contemp. Math. 53, Amer. Math. Soc., Providence, RI, 1986, pp. 207–214

    1984

  16. [24]

    Hoffstein and M

    J. Hoffstein and M. Rosen. Average values of L-series in function fields . J. Reine Angew. Math. 426 (1992), 117–150. MULTIPLE DIRICHLET SERIES PREDICTIONS FOR MOMENTS OF L-FUNCTIONS 25

    1992

  17. [25]

    Iwaniec and E

    H. Iwaniec and E. Kowalski. Analytic number theory , volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence , Rhode Island, 2004

    2004

  18. [26]

    M. Jutila. On the mean value of L( 1 2 , χ ) for real characters . Analysis 1 (2) (1981), 149–161

    1981

  19. [27]

    Katz and P

    N. Katz and P. Sarnak. Zeroes of zeta functions and symmetry . Bull. Amer. Math. Soc. 36 (1) (1999), 1–26

    1999

  20. [28]

    J. P. Keating and N. C. Snaith. Random matrix theory and ζ(1/ 2+ it). Comm. Math. Phys. 214 (2000), 57–89

    2000

  21. [29]

    J. P. Keating and N. C. Snaith. Random matrix theory and L-functions at s = 1/ 2. Comm. Math. Phys. 214 (2000), 91–100

    2000

  22. [30]

    Keating and N.C

    J.P. Keating and N.C. Snaith. Random matrices and L-functions . J. Phys. A: Math. Gen. 36 (2003) 2859–2881

    2003

  23. [31]

    Radziwi/suppress l/suppress l and K

    M. Radziwi/suppress l/suppress l and K. Soundararajan.Moments and distribution of central L-values of quadratic twists of elliptic curves . Invent. Math. 202 (2015), 1029–1068

    2015

  24. [32]

    The first moment of quadratic twists of modular L-functions

    Quanli Shen. The first moment of quadratic twists of modular L-functions. Acta Arith. 206 (2022), 313–337

    2022

  25. [33]

    Soundararajan

    K. Soundararajan. Nonvanishing of quadratic Dirichlet L-functions at s = 1

  26. [34]

    Ann. of Math. (2) 152 (2) (2000), 447–488

    2000

  27. [35]

    Soundararajan and M

    K. Soundararajan and M. P. Young. The second moment of quadratic twists of modular L-functions. J. Eur. Math. Soc. 12 (2010), 1097–1116

    2010

  28. [36]

    Taylor and A

    R. Taylor and A. Wiles. Ring-theoretic properties of certain Hecke algebras . Ann. of Math. 141 (3) (1995), 553–572

    1995

  29. [37]

    A. Wiles. Modular elliptic curves and Fermat’s last theorem . Ann. of Math. 141 (3) (1995), 443–551

    1995

  30. [38]

    M. P. Young. The third moment of quadratic Dirichlet L-functions. Selecta Math. (N.S.) 19 (2) (2013), 509–543

    2013

  31. [39]

    Q. Zhang. Integral mean values of modular L-functions. J. Number Theory 115 (2005), 100–122

    2005

  32. [40]

    Q. Zhang. Integral mean values of Maass L-functions. Int. Math. Res. Not. 2006 (2006), 1–19

    2006

  33. [41]

    Q. Zhang. Applications of multiple Dirichlet series in mean values of L-functions. In: Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory (Bretton Woods, New Hampshire, 2005), Proc. Sympos. Pure Math. 75, Amer. Math. Soc., Providence, RI, 2006, pp. 43–57....

    2005

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