Distribution of values of higher derivatives of $L'(s,\chi)/L(s,\chi)$

Samprit Ghosh

REVIEW 3 major objections 5 minor 6 references

Distribution of values of higher derivatives of $L'(s,\chi)/L(s,\chi)$

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For Re(s)>1, the first derivative of L'/L(s,χ) has a limiting probability density over Dirichlet characters.

desk verdict Plausible extension of Ihara's theory to (L'/L)', but the proof's convergence step has a load-bearing gap; the m=2 obstruction is the cleanest part. read the letter →

arxiv 2601.05189 v2 pith:OXLDE2FZ submitted 2026-01-08 math.NT

Samprit Ghosh This is my paper · ORCID

classification math.NT MSC 11M0611M41

keywords DirichletL-functionsvaluedistributionM-functionslogarithmicderivativeprobabilitydensityBellpolynomialshigherderivativesprimeconductors

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 that for any s with real part σ>1, the values of the derivative of L'/L(s,χ) — where χ runs over Dirichlet characters with prime conductor — converge, on average, to a smooth probability density M_{σ,1}(w) on the complex plane. This extends to the first derivative the known distribution law for L'/L itself. The density is built by convolving explicit local densities attached to each prime, using the uniform distribution of the character values on the circle. For the second derivative, the same construction only works when σ>2.93, and the paper explains why higher derivatives require progressively larger σ.

What carries the argument

The central object is the 'M-function' M_{σ,P}, a probability density on C built by convolution of per-prime densities M_{σ,℘}. For a single prime ℘, M_{σ,℘} is obtained by a Jacobian change of variables from the uniform measure on the unit circle t↦tN_℘^{-σ} through the map w=A t/(1-t)^2, yielding a distribution supported on the image curve. For higher derivatives the local map becomes w_m(z) = (−log N_℘)^{m+1} times a rational function involving Bell polynomials; the injectivity of this map on the disc controls the existence of a density.

What would settle it

Take σ=2, compute the empirical distribution of L'(s,χ) over all prime-conductor characters up to X; if the histogram does not converge to a smooth density as X grows, the theorem is wrong. Alternatively, find a sequence of points w_P in the support of M_{σ,P} with M_{σ,P}(w_P)→∞ as |P|→∞, which would directly contradict the boundedness claim needed for uniform convergence.

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Extended reading notes

Core claim

Theorem 3.7 states that for any s with Re(s)>1 there is a real-valued, nonnegative, C∞ function M_{σ,1} on C with unit integral such that Avg_χ Φ(L'(s,χ)) = ∫_C M_{σ,1}(w)Φ(w)|dw| for every continuous Φ, where L(s,χ)=L'(s,χ)/L(s,χ). The proof approximates L' by its finite-prime truncation L'_P, applies a change of variables that turns the uniform measure on the torus into a density on a curve, and convolves over primes; the passage to the infinite product rests on a claimed uniform convergence of the densities M_{σ,P}. For m≥2, the paper shows the analogous local function w_m(z) fails to be one-to-one on the unit disc, and for m=2 a density can be defined only when σ>2.93.

Load-bearing premise

The assertion that M_{σ,P} is bounded in w uniformly in P, used to pass from finite-prime densities to the limiting density, is not implied by the stated properties and is in fact false for a single prime, where M_{σ,℘} is a delta distribution on a curve.

Editorial extensions

If this is right

If Theorem 3.7 holds, the Fourier transform of M_{σ,1} equals the average of exp(i Re(z L'(s,χ))), giving explicit characteristic-function formulas for the distribution.
The support of M_{σ,1} is bounded for σ>1, so all moments exist and the density has an entire Fourier transform.
For the second derivative, the existence of a density only for σ>2.93 implies that value distribution for higher derivatives becomes harder as the order grows, requiring stronger conditions on σ.
The formula for the first derivative recovers the earlier moment computations for (a,b)-th moments at s=1 as a special case.

Reading between the lines

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

Editorial: The uniform convergence step in Proposition 3.5 is not fully justified: nonnegativity and unit integral do not imply local boundedness, and for one prime the density M_{σ,℘} is a distribution, so the claimed limit may only exist as a measure.
Editorial: The same change-of-variable technique could be applied to the non-injective higher-derivative maps by integrating over the preimage set with multiplicity, which might produce densities for all σ>1 at the price of a combinatorial multiplicity factor.
Editorial: The threshold σ>2.93 for m=2 is tied to the smallest zero of the numerator of w'_2(z); computing analogous zeros for m≥3 would give explicit thresholds and possibly a general bound σ_m ≫ log m.
Editorial: Because the support is compact for σ>1, the density M_{σ,1} can in principle be recovered numerically from finitely many moments, offering a check of the theorem.

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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

The paper extends Ihara's M-function construction to the first derivative of the logarithmic derivative of Dirichlet L-functions. That functional is new in this setting, and the local computations—especially the Jacobian for the |P|=1 case—are competently done. The discussion of why the method fails for m≥2 is also genuinely informative: the explicit m=2 obstruction, with the zero of w'_2 inside the unit disc forcing σ>2.93, is the kind of concrete observation one wants to see. I credit the author for being honest about where the method breaks down.

The soft spots are serious, though. The stress-test note is correct: Proposition 3.5 claims M_{σ,P} is bounded from nonnegativity and unit integral, which is false. For |P|=1 the object is a delta distribution on a curve, not a function. The displayed inequality bounding the difference M_{σ,P∪{℘}}−M_{σ,P} silently assumes a uniform L∞ bound, so the induction has no valid base case. Without a different argument, the uniform convergence to M_σ is not established, and Theorem 3.7—and with it the main Theorem 1.5—does not follow. The C∞ claim is also asserted without proof; one would need to show the infinite convolution of singular measures produces a smooth density, which is plausible but not shown here. Minor issues: the citation to Lemma 4.1 in the proof of Theorem 3.7 should be Lemma 3.3, and the use of [2, Theorem 1.2] to fix the total mass seems mismatched, since those moment formulas are at s=1, not at general σ>1.

The paper is not ready as is. The main theorem is unproven, and the flaws are load-bearing rather than cosmetic. But the underlying idea is recognizable and likely repairable—one could try to prove convergence in distribution plus some smoothing argument instead of uniform L∞ bounds. The m=2 obstruction is a nice standalone observation.

I would not cite the main theorem yet. The obstruction might be worth citing after independent verification, but the current form is not a reliable source. That said, a serious referee should see it; the issues are fixable and the topic is of interest to the value-distribution community. I'd send it to review with clear guidance to focus on the convergence step.

Referee Report

3 major / 5 minor

Summary. The paper studies the value distribution of the derivative of L'/L(s,χ) for Dirichlet characters, aiming to prove an analogue of Ihara's M-function theorem. For Re(s)>1, the author claims existence of a C∞ probability density M_{σ,1} on C such that averages over characters of Φ(L'(s,χ)) equal ∫ M_{σ,1} Φ. The method is to use Ihara's uniform distribution lemma, define finite-prime densities M_{σ,P} via change of variables and convolutions, and then let P tend to all primes using a uniform convergence argument. The paper also computes the form of local factors for higher derivatives with Faà di Bruno's formula and shows that for the second derivative the local map is non-injective unless σ is large (σ>2.93), indicating an obstruction to the same method.

Significance. If Theorem 1.5 were proved, it would be a natural and worthwhile extension of Ihara's distribution theorem to the second logarithmic derivative, with a concrete probabilistic interpretation. The paper is clearly written, and the obstruction analysis for higher derivatives is informative and honest about the limitations of the method. The use of Ihara's uniform distribution lemma is appropriate. However, the proof contains a serious gap: the finite-prime objects M_{σ,P} are not genuine functions for |P|=1, and the uniform convergence step used to pass to the limit is invalid as written. The main theorem is therefore not established in the current version.

major comments (3)

[§3, Theorem 3.4, Eq. (6)] For |P|=1, the proposed density M_{σ,℘} is defined with a Dirac delta factor δ(r-N_℘^{-σ}) and is therefore a singular measure, not a real-valued function on C. Theorem 3.4 nevertheless asserts M_{σ,P}:C→R satisfying pointwise properties (1)–(3). This is internally inconsistent. Consequently, the convolution definition (7) does not produce a function for finite P, and all subsequent pointwise operations on M_{σ,P} are undefined. This is not a minor technicality: the proof of Proposition 3.5 and the statement of Theorem 3.7 rely on these objects being functions.
[§3, Proposition 3.5] The proof asserts 'by (1) and (3) of Theorem 3.4, M_{σ,P} is bounded.' This inference is invalid: nonnegativity and unit total integral do not imply L∞ boundedness. For |P|=1, the object in question is a delta distribution, which is not even a function. The displayed bound for |M_{σ,P∪{℘}} - M_{σ,P}| therefore has no valid base case, and the claimed uniform convergence of M_{σ,P_y} to a continuous M_σ is not established. Since Theorem 3.7 and Theorem 1.5 depend on this convergence, the main theorem is unproven. A different argument (e.g., via characteristic functions/Fourier inversion) is needed.
[§3, Theorem 3.7; Theorem 1.5] Even if the convergence asserted in Proposition 3.5 were repaired, it would only yield a continuous limit M_σ. Theorem 1.5 claims that M_{σ,1} is C∞, and Proposition 3.5 gives no derivative estimates or smoothing argument. No separate proof of infinite differentiability is supplied. The C∞ property is therefore unsupported as the manuscript stands.

minor comments (5)

[Title/Abstract] The title contains typos ('DISTIBUTION', 'V ALUES'); please correct.
[§3, Eq. (6)] The coefficient of δ(r-N_℘^{-σ}) appears to be off by a factor of (log N_℘)^2 relative to J^{-1} computed immediately above; please check the normalization.
[§3, Theorem 3.4, property (2)] Property (2) reads 'M_{σ,P}(w)=M_{σ,P}(w)', which is tautological; presumably a symmetry such as M_{σ,P}(w)=overline{M_{σ,P}(\bar w)} was intended.
[§3, Proposition 3.5] The phrase 'for σ>1/2 (in fact, 1/4)' is ambiguous. If the argument is based on the convergence of ∑ N^{-4σ}, the condition is σ>1/4, not σ>1/2.
[§4.1, m=2 case] The condition that w'_2 has no zeros inside the disc of radius ρ ensures local injectivity but not, by itself, the global one-to-one change of variables needed to produce a density. Additional justification is needed for the claim that M_{σ,2} exists for σ>2.93.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the construction is a genuine change-of-variables/convolution argument, not a reduction to its inputs; the flagged issue is a proof gap, not a circular step.

full rationale

The derivation of Theorem 1.5 runs: Lemma 3.1 (Ihara's uniform distribution on T_P) is external; the map g_{σ,P} is a concrete change of variables; M_{σ,P} is defined by the resulting Jacobian/convolution; the character average is equated to ∫ M_{σ,P} Φ by Lemma 3.1, not by assuming the target density. No parameter is fitted to the target, and no uniqueness theorem is imported from the author's own work. The only author self-citation is the note after Theorem 3.7 identifying the total mass ∫M_σ with the zero-th moment μ^{(0,0)}(1) from [2]; this is not load-bearing because the distributional identity is already proved from Lemma 3.1 and Prop. 3.5, and the zero-th moment is 1 by definition (it is also available from the bounded-support argument in §4.2). Section 4.2's admission that higher derivatives currently fail is a limitation, not circularity. The real weakness is Prop. 3.5: 'by (1) and (3) of Theorem 3.4, M_{σ,P} is bounded' is invalid (for |P|=1, M_{σ,℘} is a delta distribution), so the uniform convergence to M_σ is not established. That is a serious correctness gap, but it is an unsupported inference, not a reduction of the theorem to its own premises.

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

The central derivation imports Ihara's equidistribution lemma and Jessen-Wintner theory; there are no fitted numerical constants. The two ad hoc assumptions listed above are where the proof's load-bearing fragility lives, and the first of them is demonstrably invalid.

assumptions (5)

domain assumption Lemma 3.1 (Ihara): the tuples (χ(℘))_{℘∈P} are uniformly distributed on the torus T_P for the family of characters.
This equidistribution statement is the engine of the whole argument; it is quoted from Ihara [3, Lemma 4.3.1] and not re-proven here.
standard math Jessen-Wintner distribution-function framework, including weak convergence criteria and convolution definitions.
Used in Section 2 to convert averages over characters into integrals against distribution functions; standard external background.
ad hoc to paper M_{σ,P} is bounded because it is nonnegative and integrates to 1 (used in the proof of Proposition 3.5).
This premise is false as stated: a nonnegative L^1 function need not be bounded, and for |P|=1 the object is a delta distribution. It is load-bearing for the uniform convergence of M_{σ,P}.
ad hoc to paper For m=2, absence of zeros of w'_2 inside the disc of radius ρ is sufficient for the change-of-variables to produce a valid density.
Section 4.1 checks the location of one zero of w'_2 and then asserts the density exists for σ>2.93, but no proof of global injectivity or of the induced density and its convergence over P_y is given.
standard math The Jacobian for w(z)=A z/(1-z)^2 (Equation 5) is correct.
This Jacobian is stated to have been computed using a computer algebra system; it is in fact correct by direct computation, so this is not a source of error, but the paper does not show the derivation.

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Cite this review

Pith. "Pith review of Distribution of values of higher derivatives of $L'(s,\chi)/L(s,\chi)$." pith.science (2026). https://pith.science/paper/OXLDE2FZ

@misc{pith2026260105189,
  author       = {Pith},
  title        = {Pith review of: Distribution of values of higher derivatives of $L'(s,\chi)/L(s,\chi)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OXLDE2FZ}},
  note         = {Machine review of arXiv:2601.05189}
}

abstract

In this article, we study the value distribution theory for the first derivative of the logarithmic derivative of Dirichlet $L$-functions, generalizing certain results of Ihara, Matsumoto et al. related to ``$M$-functions'' for $\sigma = \operatorname{Re}(s) > 1$. We then discuss the main obstruction toward generalization to higher derivatives.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 1 linked inside Pith

[1]

17946, 2024, (to appear in International Journal of Number Theory)

Samprit Ghosh,Higher Euler-Kronecker Constants of Number fields, https://arxiv.org/abs/2411. 17946, 2024, (to appear in International Journal of Number Theory)

2024
[2]

org/abs/2509.06390, 2025, arXiv

,Moments of higher derivatives of the logarithmic derivative of Dirichlet L-functions, https://arxiv. org/abs/2509.06390, 2025, arXiv

arXiv 2025
[3]

M-functions

Y. Ihara,On “M-functions” closely related to the distribution of L’/L-values, RIMS, Kyoto University,44, page 893-954 (2008)

2008
[4]

M-functions

Yasutaka Ihara and Kohji Matsumoto,On “M-functions” closely related to the distribution of L’/L-values, Moscow Mathematical Journal, Vol 11, issue 1 (2011), 73–111

2011
[5]

Jessen and A

B. Jessen and A. Wintner,Distribution Functions and the Riemann Zeta Function, Trans. Amer. Math. Soc.,38, No. 1, (1935), 48–88

1935
[6]

Murty,Distribution of values of L′/L(σ, χD), Moscow Mathematical Journal, Vol 15, issue 3 (2015), 497–509

Mariam Mourtada and Kumar V . Murty,Distribution of values of L′/L(σ, χD), Moscow Mathematical Journal, Vol 15, issue 3 (2015), 497–509. MATHEMATICALSCIENCES468, UNIVERSITY OFCALGARY, 2500 UNIVERSITYDRIVENW, CALGARY, ALBERTA, T2N 1N4, CANADA. Email address:samprit.ghosh@ucalgary.ca 11

2015

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

17946, 2024, (to appear in International Journal of Number Theory)

Samprit Ghosh,Higher Euler-Kronecker Constants of Number fields, https://arxiv.org/abs/2411. 17946, 2024, (to appear in International Journal of Number Theory)

2024

[2] [2]

org/abs/2509.06390, 2025, arXiv

,Moments of higher derivatives of the logarithmic derivative of Dirichlet L-functions, https://arxiv. org/abs/2509.06390, 2025, arXiv

arXiv 2025

[3] [3]

M-functions

Y. Ihara,On “M-functions” closely related to the distribution of L’/L-values, RIMS, Kyoto University,44, page 893-954 (2008)

2008

[4] [4]

M-functions

Yasutaka Ihara and Kohji Matsumoto,On “M-functions” closely related to the distribution of L’/L-values, Moscow Mathematical Journal, Vol 11, issue 1 (2011), 73–111

2011

[5] [5]

Jessen and A

B. Jessen and A. Wintner,Distribution Functions and the Riemann Zeta Function, Trans. Amer. Math. Soc.,38, No. 1, (1935), 48–88

1935

[6] [6]

Murty,Distribution of values of L′/L(σ, χD), Moscow Mathematical Journal, Vol 15, issue 3 (2015), 497–509

Mariam Mourtada and Kumar V . Murty,Distribution of values of L′/L(σ, χD), Moscow Mathematical Journal, Vol 15, issue 3 (2015), 497–509. MATHEMATICALSCIENCES468, UNIVERSITY OFCALGARY, 2500 UNIVERSITYDRIVENW, CALGARY, ALBERTA, T2N 1N4, CANADA. Email address:samprit.ghosh@ucalgary.ca 11

2015