Joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives

Shengbo Zhao

arxiv: 2604.04510 · v1 · submitted 2026-04-06 · 🧮 math.NT

Joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives

Shengbo Zhao This is my paper

Pith reviewed 2026-05-10 20:16 UTC · model grok-4.3

classification 🧮 math.NT

keywords Dirichlet L-functionsextreme valueslogarithmic derivativesresonance methodjoint valuescritical lineanalytic number theory

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The pith

Dirichlet L-functions and their logarithmic derivatives achieve joint extreme values

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

This paper proves that Dirichlet L-functions and their logarithmic derivatives can take on large values at the same points using the resonance method. The work extends earlier results that handled these functions separately. Readers interested in the analytic properties of L-functions would find this relevant because joint extremes reveal more about their collective behavior near the critical line. The resonance technique constructs auxiliary functions that amplify both quantities together.

Core claim

We establish joint extreme values of Dirichlet L-functions and their logarithmic derivatives using the resonance method. Our results extend previous work of Aistleitner et al. (2019) and Yang (2023).

What carries the argument

The resonance method, which constructs points where both the L-function and its logarithmic derivative are forced to large values simultaneously

If this is right

Joint extreme values occur for an L-function and its logarithmic derivative at the same points where the resonance method applies.
The technical conditions sufficient for single-function extremes carry over directly to the paired case.
The maximal orders of growth for the pair are achieved together rather than in conflict.

Where Pith is reading between the lines

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

The same resonance construction might extend without change to triples consisting of an L-function, its first derivative, and its second derivative.
Explicit computation of resonance points for the first few characters could produce concrete numerical examples of the joint extremes.
The result implies the logarithmic derivative does not systematically cancel the large values produced by resonance on the L-function itself.

Load-bearing premise

The resonance method extends without new obstructions to the joint setting of an L-function and its logarithmic derivative, relying on the technical conditions already present in the cited prior works.

What would settle it

A numerical search for a small-modulus Dirichlet character and a point near the critical line where the predicted joint lower bounds for both the L-function and its log derivative fail to hold simultaneously would falsify the existence claim.

read the original abstract

In this paper, we establish joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives using the resonance method. Our results extend previous work of Aistleitner et al. (2019) and Yang (2023).

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.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to establish joint extreme values of Dirichlet L-functions and their logarithmic derivatives on the critical line by adapting the resonance method. It constructs a resonator that simultaneously amplifies both L(1/2+it) and (L'/L)(1/2+it), deriving the necessary moment estimates and correlation bounds from Euler-product representations and the technical hypotheses of Aistleitner et al. (2019) and Yang (2023), thereby extending those individual extreme-value results to the joint setting.

Significance. If the adaptation holds, the result is a meaningful technical extension that demonstrates the resonance method carries over to joint distributions without new arithmetic obstructions or additional input. This strengthens the framework for studying extreme values of L-functions by incorporating their derivatives, offering a more complete picture of their joint behavior. The paper receives credit for deriving all required bounds from existing Euler-product estimates and prior conditions rather than introducing new parameters or hypotheses.

minor comments (2)

The abstract is concise but does not state the precise form of the joint extreme-value theorems (e.g., the range of the joint distribution or the explicit resonator construction); a slightly expanded abstract would improve readability without altering the technical content.
Notation for the joint resonator and the correlation bounds in the main argument could be cross-referenced more explicitly to the corresponding statements in Aistleitner et al. (2019) and Yang (2023) to highlight exactly which hypotheses are reused.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies the resonance method to obtain joint extreme values for L-functions and their logarithmic derivatives. All moment estimates, correlation bounds, and technical hypotheses are taken directly from the externally cited works of Aistleitner et al. (2019) and Yang (2023), which are independent of the present manuscript. No parameters are fitted to the target data and then re-labeled as predictions, no self-definitional loops appear in the derivation, and the central claim does not reduce to a renaming or to a load-bearing self-citation chain. The argument therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions from analytic number theory for the existence and analytic continuation of Dirichlet L-functions and on the technical setup of the resonance method as developed in the cited works; no new free parameters or invented entities are visible from the abstract.

axioms (2)

domain assumption Standard analytic continuation and functional equation properties of Dirichlet L-functions
Invoked implicitly when applying the resonance method to these functions.
domain assumption Technical conditions of the resonance method as established in Aistleitner et al. (2019) and Yang (2023)
The extension claim rests on these prior setups carrying over to the joint case.

pith-pipeline@v0.9.0 · 5312 in / 1256 out tokens · 47202 ms · 2026-05-10T20:16:44.028433+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We establish joint extreme values of Dirichlet L-functions and their logarithmic derivatives using the resonance method... Theorems 1.1–1.4 and the resonator R(χ) = ∏_{p≤X} (1−r(p)χ(p))^{-1}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

S2/S1 ≥ e^{ℓγ} (log X)^ℓ − … and the explicit form of S(σ,ℓ) and H(σ,ℓ)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Aistleitner, Lower bounds for the maximum of the Riemann zeta function along vertical lines, Math

[1]C. Aistleitner, Lower bounds for the maximum of the Riemann zeta function along vertical lines, Math. Ann., 365 (2016), pp. 473–496. [2]C. Aistleitner, K. Mahatab, and M. Munsch, Extreme valuesof the Riemann zeta function on the 1-line, Int. Math. Res. Not., IMRN 2019 (2019), pp. 6924–6932. [3]C. Aistleitner, K. Mahatab, M. Munsch, and A. Peyrot, On la...

work page 2016
[2]

Bondarenko and K

[5]A. Bondarenko and K. Seip, Large greatest common divisor sums and extreme values of the Riemann zeta function, Duke Math. J., 166 (2017), pp. 1685–1701. [6]A. Bondarenko and K. Seip, Extreme values of the Riemann zeta function and its argument, Math. Ann., 372 (2018), pp. 999–1015. [7]A. Chirre, Extreme values for Sn(σ, t) near the critical line, J. Nu...

work page 2017
[3]

[16]H. L. Montgomery, Extreme values of the Riemann zeta function, Comment. Math. Helv., 52 (1977), pp. 511—-518. [17]M. R. Mourtada and V. K. Murty, Omega theorems for L′(1, χD), Int. J. Number Theory, 9 (2013), pp. 561—-581. [18]J. B. Rosser and L. Schoenfeld, Approximateformulasfor some functions of prime numbers, Illinois J. Math., 6 (1962), pp. 64–94...

work page 1977
[4]

[20]S. M. Voronin, Lower bounds in Riemann zeta-function theory, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), pp. 882–892,

work page 1988
[5]

[21]X. M. W. and Y. D., Extreme values of Dirichlet polynomials with multiplicative coefficients, J. Number Theory, 258 (2024), pp. 173–180. [22]X. Xiao and Q. Yang, A note on large values of L(σ, χ), Bull. Aust. Math. Soc., 105 (2022), pp. 412—-418. [23]D. Yang, Extreme values of derivatives of zeta and L-functions, Bull. Lond. Math. Soc., 56 (2023), pp....

work page arXiv 2024

[1] [1]

Aistleitner, Lower bounds for the maximum of the Riemann zeta function along vertical lines, Math

[1]C. Aistleitner, Lower bounds for the maximum of the Riemann zeta function along vertical lines, Math. Ann., 365 (2016), pp. 473–496. [2]C. Aistleitner, K. Mahatab, and M. Munsch, Extreme valuesof the Riemann zeta function on the 1-line, Int. Math. Res. Not., IMRN 2019 (2019), pp. 6924–6932. [3]C. Aistleitner, K. Mahatab, M. Munsch, and A. Peyrot, On la...

work page 2016

[2] [2]

Bondarenko and K

[5]A. Bondarenko and K. Seip, Large greatest common divisor sums and extreme values of the Riemann zeta function, Duke Math. J., 166 (2017), pp. 1685–1701. [6]A. Bondarenko and K. Seip, Extreme values of the Riemann zeta function and its argument, Math. Ann., 372 (2018), pp. 999–1015. [7]A. Chirre, Extreme values for Sn(σ, t) near the critical line, J. Nu...

work page 2017

[3] [3]

[16]H. L. Montgomery, Extreme values of the Riemann zeta function, Comment. Math. Helv., 52 (1977), pp. 511—-518. [17]M. R. Mourtada and V. K. Murty, Omega theorems for L′(1, χD), Int. J. Number Theory, 9 (2013), pp. 561—-581. [18]J. B. Rosser and L. Schoenfeld, Approximateformulasfor some functions of prime numbers, Illinois J. Math., 6 (1962), pp. 64–94...

work page 1977

[4] [4]

[20]S. M. Voronin, Lower bounds in Riemann zeta-function theory, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), pp. 882–892,

work page 1988

[5] [5]

[21]X. M. W. and Y. D., Extreme values of Dirichlet polynomials with multiplicative coefficients, J. Number Theory, 258 (2024), pp. 173–180. [22]X. Xiao and Q. Yang, A note on large values of L(σ, χ), Bull. Aust. Math. Soc., 105 (2022), pp. 412—-418. [23]D. Yang, Extreme values of derivatives of zeta and L-functions, Bull. Lond. Math. Soc., 56 (2023), pp....

work page arXiv 2024