Mean values of derivatives of $L$-functions in function fields: IV

Hwanyup Jung; Julio Andrade

arxiv: 1906.10373 · v1 · pith:SCQL5B3Lnew · submitted 2019-06-25 · 🧮 math.NT

Mean values of derivatives of L-functions in function fields: IV

Julio Andrade , Hwanyup Jung This is my paper

Pith reviewed 2026-05-25 16:43 UTC · model grok-4.3

classification 🧮 math.NT

keywords mean valuesderivativesDirichlet L-functionsfunction fieldsquadratic charactersasymptotic formulaeapproximate functional equation

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The pith

The mean value of the μ-th derivative of quadratic Dirichlet L-functions over the rational function field is given by an explicit polynomial in the degree whose coefficients display arithmetic dependence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes, for every fixed positive integer μ, the average size of the μ-th derivative of quadratic Dirichlet L-functions attached to the rational function field. It produces the complete asymptotic expansion as a polynomial in the degree parameter n of the character, with every coefficient written out. The expansion makes visible how lower-order terms depend on the arithmetic of the underlying finite field and its extensions. A sympathetic reader would care because these averages control the typical magnitude of L-values and their derivatives, which govern questions about zero distributions and value distributions in the function-field setting.

Core claim

For each integer μ ≥ 1 the mean value of the μ-th derivative of a quadratic Dirichlet L-function over F_q(t) equals an explicit polynomial in the degree n whose leading term is of size roughly n^{2μ+1} and whose remaining coefficients are expressed through sums that encode the arithmetic of the function field.

What carries the argument

The analogue of the approximate functional equation for these L-functions together with the Riemann hypothesis for curves over finite fields.

If this is right

The leading coefficient of the mean value is determined by a constant depending only on μ and q.
Every lower-degree term in the polynomial is nonzero and carries explicit dependence on the characteristic and extension degrees.
The result recovers and extends the earlier cases for the undifferentiated L-function and its first derivative.
The arithmetic sums appearing in the coefficients can be evaluated in closed form for any fixed q.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same method should apply directly to mean values of derivatives at points other than the central point.
The explicit polynomials supply concrete test cases for any conjectural moment formulae that are later transplanted from function fields to number fields.
One can check the formulae by enumerating all quadratic characters of small degree over F_2(t) or F_3(t) and computing the derivatives numerically.

Load-bearing premise

The approximate functional equation holds in exact form for quadratic Dirichlet L-functions in the function field setting.

What would settle it

A direct numerical computation of the mean value for fixed small μ and small n over a small finite field q that disagrees with the predicted polynomial coefficients.

read the original abstract

In this series, we investigate the calculation of mean values of derivatives of Dirichlet $L$-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For $\mu\geq1$ an integer, we compute the mean value of the $\mu$-th derivative of quadratic Dirichlet $L$-functions over the rational function field. We obtain the full polynomial in the asymptotic formulae for these mean values where we can see the arithmetic dependence of the lower order terms that appears in the asymptotic expansion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Straightforward extension of the series to full polynomials for arbitrary-order derivatives, using the usual function-field toolkit.

read the letter

The main thing here is that the authors push the earlier work in the series from leading-term asymptotics to the complete polynomial in the mean value of the μ-th derivative, for any fixed μ. They keep the same setup over the rational function field and extract the arithmetic factors that show up in the lower-order coefficients. That matches what the abstract claims and what the stress-test note flags as the incremental step. The tools are the standard ones: the analogue of the approximate functional equation plus Weil's theorem, both unconditional in this setting. No sign that the derivation introduces new circularity or fitting; it treats the prior papers as inputs and builds the higher-derivative case on top. The arithmetic dependence in the lower terms is expected once you have the Euler-product structure, so that part feels natural rather than surprising. The work stays within an established program, so the novelty is mainly technical completeness rather than a new direction. Soft spots are minor: the paper is incremental by design, and anyone wanting to use the result will still need to check the details of how the higher derivatives are handled in the approximate functional equation. Nothing in the abstract or the reader's summary suggests a load-bearing gap. This is the kind of paper that belongs in a specialist journal on analytic number theory over function fields. Readers already following the series will find it useful for the explicit polynomials; outsiders probably won't. It is solid enough to deserve referee time rather than a desk reject, even if the revisions are light. I would bring it to a reading group only if the group is already working on function-field moments.

Referee Report

0 major / 1 minor

Summary. The paper generalizes prior results in the series to compute the mean value of the μ-th derivative (μ ≥ 1 integer) of quadratic Dirichlet L-functions over the rational function field. It derives the full asymptotic polynomial using the analogue of the approximate functional equation together with Weil's theorem (RH for curves over finite fields), making the arithmetic dependence of lower-order terms explicit.

Significance. If the derivations hold, the work supplies unconditional, explicit asymptotics for higher derivatives in the function-field setting, with visible arithmetic factors arising from the Euler product. This strengthens the series by extending the range of moments treated and provides a clean test case for phenomena expected in the number-field analogues.

minor comments (1)

The abstract refers to 'the rational function field' without specifying the constant field F_q; a brief parenthetical on the dependence on q would improve clarity for readers outside the series.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper and for recommending acceptance. The report correctly identifies the main contribution as the derivation of the complete asymptotic polynomial for the mean values of higher derivatives using the approximate functional equation and Weil's theorem.

Circularity Check

0 steps flagged

No significant circularity; relies on unconditional tools and prior independent results

full rationale

The derivation generalizes the first paper in the series via the standard analogue of the approximate functional equation (which follows directly from the functional equation of the L-functions) together with Weil's theorem on the Riemann hypothesis for curves over finite fields. Both ingredients are unconditional in the function-field setting and are not derived within this paper. The central asymptotic polynomial for the μ-th derivative mean value is obtained by direct computation from these inputs; no equation reduces the claimed result to a fitted parameter or to a self-citation chain by construction. Reliance on the earlier papers in the series is treated as independent input rather than a load-bearing self-definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on two standard results in function-field arithmetic whose proofs predate the paper.

axioms (2)

standard math Riemann hypothesis for curves over finite fields (Weil's theorem)
Invoked explicitly in the abstract to evaluate the mean values.
domain assumption Analogue of the approximate functional equation for Dirichlet L-functions over function fields
Stated in the abstract as the main computational tool.

pith-pipeline@v0.9.0 · 5620 in / 1222 out tokens · 23766 ms · 2026-05-25T16:43:38.457206+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Andrade, Mean values of derivatives of L-functions in function ﬁelds: II , J

J. Andrade, Mean values of derivatives of L-functions in function ﬁelds: II , J. Number Theory 183 (2018), 24–39

work page 2018
[2]

, Mean values of derivatives of L-functions in function ﬁelds: III , to appear in the Proc. Roy. Soc. Edinburgh Sect. A (2017) (2017)

work page 2017
[3]

, A simple proof of the mean value of |K2(O)| in function ﬁelds , C. R. Math. Acad. Sci. Paris 353 (2015), no. 8, 677–682

work page 2015
[4]

37 (2016), 311–327

, Rudnick and Soundararajan ’s theorem for function ﬁelds , Finite Fields Appl. 37 (2016), 311–327

work page 2016
[5]

J. C. Andrade and J. P. Keating, The mean value of L( 1 2 , χ) in the hyperelliptic ensemble , J. Number Theory 132 (2012), no. 12, 2793–2816

work page 2012
[6]

Andrade and S

J. Andrade and S. Rajagopal, Mean values of derivatives of L-functions in function ﬁelds: I , J. Math. Anal. Appl. 443 (2016), no. 1, 526–541

work page 2016
[7]

J. B. Conrey, The fourth moment of derivatives of the Riemann zeta-functi on, Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 153, 21–36

work page 1988
[8]

J. B. Conrey, M. O. Rubinstein, and N. C. Snaith, Moments of the derivative of characteristic polynomials wi th an application to the Riemann zeta function , Comm. Math. Phys. 267 (2006), no. 3, 611–629. MEAN V ALUES OF L-FUNCTIONS AND ITS DERIV ATIVES: IV 15

work page 2006
[9]

Faifman and Z

D. Faifman and Z. Rudnick, Statistics of the zeros of zeta functions in families of hype relliptic curves over a ﬁnite ﬁeld , Compos. Math. 146 (2010), no. 1, 81–101

work page 2010
[10]

A. M. Florea, Improving the error term in the mean value of L( 1 2 , χ) in the hyperelliptic ensemble , Int. Math. Res. Not. IMRN 20 (2017), 6119–6148

work page 2017
[11]

S. M. Gonek, Mean values of the Riemann zeta function and its derivatives , Invent. Math. 75 (1984), no. 1, 123–141

work page 1984
[12]

Hoﬀstein and M

J. Hoﬀstein and M. Rosen, Average values of L-series in function ﬁelds , J. Reine Angew. Math. 426 (1992), 117–150

work page 1992
[13]

A. E. Ingham, Mean-Value Theorems in the Theory of the Riemann Zeta-Funct ion, Proc. London Math. Soc. (2) 27 (1927), no. 4, 273–300

work page 1927
[14]

Jung, Note on the mean value of L( 1 2 , χ) in the hyperelliptic ensemble , J

H. Jung, Note on the mean value of L( 1 2 , χ) in the hyperelliptic ensemble , J. Number Theory 133 (2013), no. 8, 2706–2714

work page 2013
[15]

Jutila, On the Mean Value of L( 1 2 , χ) for Real Characters , Analysis 1 (1981), no

M. Jutila, On the Mean Value of L( 1 2 , χ) for Real Characters , Analysis 1 (1981), no. 2, 149–161

work page 1981
[16]

Rosen, Number theory in function ﬁelds , Graduate Texts in Mathematics, vol

M. Rosen, Number theory in function ﬁelds , Graduate Texts in Mathematics, vol. 210, Springer-Verlag , New York, 2002

work page 2002
[17]

D. S. Thakur, Function ﬁeld arithmetic , W orld Scientiﬁc Publishing Co., Inc., River Edge, NJ, 2004

work page 2004
[18]

W eil, Sur les courbes alg´ ebriques et les vari´ et´ es qui s’en d´ eduisent, Actualit´ es Sci

A. W eil, Sur les courbes alg´ ebriques et les vari´ et´ es qui s’en d´ eduisent, Actualit´ es Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann et Cie., Paris, 1948 (French). Department of Mathematics, University of Exeter, Exeter, EX4 4QF, United Kingdom E-mail address : j.c.andrade@exeter.ac.uk Department of Mathematics Educati...

work page 1945

[1] [1]

Andrade, Mean values of derivatives of L-functions in function ﬁelds: II , J

J. Andrade, Mean values of derivatives of L-functions in function ﬁelds: II , J. Number Theory 183 (2018), 24–39

work page 2018

[2] [2]

, Mean values of derivatives of L-functions in function ﬁelds: III , to appear in the Proc. Roy. Soc. Edinburgh Sect. A (2017) (2017)

work page 2017

[3] [3]

, A simple proof of the mean value of |K2(O)| in function ﬁelds , C. R. Math. Acad. Sci. Paris 353 (2015), no. 8, 677–682

work page 2015

[4] [4]

37 (2016), 311–327

, Rudnick and Soundararajan ’s theorem for function ﬁelds , Finite Fields Appl. 37 (2016), 311–327

work page 2016

[5] [5]

J. C. Andrade and J. P. Keating, The mean value of L( 1 2 , χ) in the hyperelliptic ensemble , J. Number Theory 132 (2012), no. 12, 2793–2816

work page 2012

[6] [6]

Andrade and S

J. Andrade and S. Rajagopal, Mean values of derivatives of L-functions in function ﬁelds: I , J. Math. Anal. Appl. 443 (2016), no. 1, 526–541

work page 2016

[7] [7]

J. B. Conrey, The fourth moment of derivatives of the Riemann zeta-functi on, Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 153, 21–36

work page 1988

[8] [8]

J. B. Conrey, M. O. Rubinstein, and N. C. Snaith, Moments of the derivative of characteristic polynomials wi th an application to the Riemann zeta function , Comm. Math. Phys. 267 (2006), no. 3, 611–629. MEAN V ALUES OF L-FUNCTIONS AND ITS DERIV ATIVES: IV 15

work page 2006

[9] [9]

Faifman and Z

D. Faifman and Z. Rudnick, Statistics of the zeros of zeta functions in families of hype relliptic curves over a ﬁnite ﬁeld , Compos. Math. 146 (2010), no. 1, 81–101

work page 2010

[10] [10]

A. M. Florea, Improving the error term in the mean value of L( 1 2 , χ) in the hyperelliptic ensemble , Int. Math. Res. Not. IMRN 20 (2017), 6119–6148

work page 2017

[11] [11]

S. M. Gonek, Mean values of the Riemann zeta function and its derivatives , Invent. Math. 75 (1984), no. 1, 123–141

work page 1984

[12] [12]

Hoﬀstein and M

J. Hoﬀstein and M. Rosen, Average values of L-series in function ﬁelds , J. Reine Angew. Math. 426 (1992), 117–150

work page 1992

[13] [13]

A. E. Ingham, Mean-Value Theorems in the Theory of the Riemann Zeta-Funct ion, Proc. London Math. Soc. (2) 27 (1927), no. 4, 273–300

work page 1927

[14] [14]

Jung, Note on the mean value of L( 1 2 , χ) in the hyperelliptic ensemble , J

H. Jung, Note on the mean value of L( 1 2 , χ) in the hyperelliptic ensemble , J. Number Theory 133 (2013), no. 8, 2706–2714

work page 2013

[15] [15]

Jutila, On the Mean Value of L( 1 2 , χ) for Real Characters , Analysis 1 (1981), no

M. Jutila, On the Mean Value of L( 1 2 , χ) for Real Characters , Analysis 1 (1981), no. 2, 149–161

work page 1981

[16] [16]

Rosen, Number theory in function ﬁelds , Graduate Texts in Mathematics, vol

M. Rosen, Number theory in function ﬁelds , Graduate Texts in Mathematics, vol. 210, Springer-Verlag , New York, 2002

work page 2002

[17] [17]

D. S. Thakur, Function ﬁeld arithmetic , W orld Scientiﬁc Publishing Co., Inc., River Edge, NJ, 2004

work page 2004

[18] [18]

W eil, Sur les courbes alg´ ebriques et les vari´ et´ es qui s’en d´ eduisent, Actualit´ es Sci

A. W eil, Sur les courbes alg´ ebriques et les vari´ et´ es qui s’en d´ eduisent, Actualit´ es Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann et Cie., Paris, 1948 (French). Department of Mathematics, University of Exeter, Exeter, EX4 4QF, United Kingdom E-mail address : j.c.andrade@exeter.ac.uk Department of Mathematics Educati...

work page 1945