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arxiv: 2605.03665 · v1 · submitted 2026-05-05 · 🧮 math.NT

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Joint extreme values of L-functions on and off the critical line

Athanasios Sourmelidis

Pith reviewed 2026-05-07 13:24 UTC · model grok-4.3

classification 🧮 math.NT
keywords L-functionscritical lineextreme valuesresonance methodHeath-Brown's methodzero-density estimatesDirichlet L-functionsSelberg class
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The pith

Any number of distinct primitive GL(1) and GL(2) L-functions can simultaneously attain large values on the critical line.

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

The paper proves that for any positive integer k, there exist points on the critical line where k distinct primitive L-functions of degree one or two simultaneously take very large values. This holds unconditionally, without assuming the Riemann hypothesis, which earlier results required once more than three functions were involved. The proof combines the resonance method with a modified version of Heath-Brown's approach to fractional moments of the zeta function, allowing avoidance of zero-distribution data for low-degree cases. Off the critical line, joint large values are obtained under zero-density estimates, recovering some prior results for Dirichlet L-functions and improving others for Selberg-class functions. A reader should care because the result shows that extreme behavior of these L-functions can be arranged jointly in a strong, hypothesis-free manner on the line of greatest arithmetic interest.

Core claim

It is shown that any number of distinct primitive GL(1) and GL(2) L-functions can simultaneously attain large values on the critical line. This is an unconditional improvement of a general result due to Heap and Li who have assumed the Riemann Hypothesis for more than three such L-functions. The joint distribution of GL(m) L-functions to the right of the critical line is also studied under certain zero-density estimates. In particular, we can partially recover results of Inoue and Li on Dirichlet L-functions and generally improve upon the work of Mahatab, Pańkowski and Vatwani on the class of L-functions introduced by Selberg. The main machinery in both cases, on and off the critical line, 1

What carries the argument

The resonance method of Soundararajan, combined with a variation of Heath-Brown's method for the fractional moments of the Riemann zeta-function that avoids zero-distribution information for L-functions of degree less than three.

Load-bearing premise

The variation of Heath-Brown's method for fractional moments works without any information on the distribution of zeros for L-functions of degree less than three.

What would settle it

A theorem or explicit calculation establishing an upper bound on the simultaneous size of log |L1(1/2+it)| + ... + log |Lk(1/2+it)| that is smaller than the claimed resonance lower bound, for some fixed k greater than or equal to four and some specific choice of primitive GL(1) or GL(2) L-functions.

read the original abstract

It is shown that any number of distinct primitive $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$-functions can simultaneously attain large values on the critical line. This is an unconditional improvement of a general result due to Heap and Li who have assumed the Riemann Hypothesis for more than three such $L$-functions. The joint distribution of $\mathrm{GL}(m)$ $L$-functions to the right of the critical line is also studied under certain zero-density estimates. In particular, we can partially recover results of Inoue and Li on Dirichlet $L$-functions and generally improve upon the work of Mahatab, Pa\'nkowski and Vatwani on the class of $L$-functions introduced by Selberg. The main machinery in both cases, on and off the critical line, is the resonance method of Soundararajan and Hilberdink/Voronin, respectively. On the critical line we additionally introduce a variation of Heath-Brown's method for the fractional moments of the Riemann zeta-function which makes it possible to avoid using any information on the zero distribution of $L$-functions whose degree is less than three.

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

2 major / 2 minor

Summary. The paper establishes that arbitrarily many distinct primitive GL(1) and GL(2) L-functions can simultaneously attain large values on the critical line, unconditionally. This improves Heap-Li, who required RH for more than three such functions. Off the critical line, the joint distribution of GL(m) L-functions is studied under zero-density estimates, partially recovering Inoue-Li for Dirichlet L-functions and improving Mahatab-Pańkowski-Vatwani for Selberg-class L-functions. The proofs rely on the resonance method (Soundararajan on the line, Hilberdink-Voronin off it) together with a new variation of Heath-Brown's method for fractional moments that avoids zero-distribution data for degree <3 L-functions.

Significance. If the variation of Heath-Brown's method is valid in the joint setting, the unconditional result on the critical line represents a clear advance, removing the RH hypothesis for arbitrarily many GL(1)/GL(2) functions. The conditional results off the line are incremental but useful. The paper supplies machine-checkable-style explicit error tracking in the resonance step and parameter-free resonance weights, which are strengths.

major comments (2)
  1. [§4] §4 (Variation of Heath-Brown's method): the central unconditional claim for arbitrarily many GL(1) and GL(2) L-functions on the critical line rests on this variation producing the required fractional-moment bounds while using no zero-distribution information for any L-function of degree <3. The exposition does not contain an explicit verification that the error terms arising from the multi-function resonance product remain controllable without implicitly invoking zero-density estimates when the product involves more than two such L-functions; a line-by-line comparison with the single-function Heath-Brown argument would clarify whether the joint case introduces new zero-free-region dependencies.
  2. [§5.3] §5.3 (Joint resonance on the critical line): the claimed improvement over Heap-Li is stated to hold for any number of functions, yet the moment bound in the multi-L-function case appears to reduce the effective range of the resonance parameter by a factor depending on the number of functions; it is not shown that this reduction remains harmless for the lower bound on the maximum when the number of L-functions grows.
minor comments (2)
  1. [§3] The notation for the joint resonance weights (e.g., the product over multiple L-functions in the definition of the mollifier) is introduced without a separate display equation; adding an explicit formula would improve readability.
  2. [References] Several citations to Heath-Brown's original fractional-moment paper are given only by author and year; page or theorem numbers would help the reader locate the precise statements being varied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the thorough review and the recommendation for major revision. We have carefully considered the comments and provide point-by-point responses below. We will revise the manuscript to incorporate clarifications as indicated.

read point-by-point responses

  1. Referee: [§4] §4 (Variation of Heath-Brown's method): the central unconditional claim for arbitrarily many GL(1) and GL(2) L-functions on the critical line rests on this variation producing the required fractional-moment bounds while using no zero-distribution information for any L-function of degree <3. The exposition does not contain an explicit verification that the error terms arising from the multi-function resonance product remain controllable without implicitly invoking zero-density estimates when the product involves more than two such L-functions; a line-by-line comparison with the single-function Heath-Brown argument would clarify whether the joint case introduces new zero-free-region dependencies.

    Authors: We appreciate the referee's suggestion for greater clarity on this point. The variation of Heath-Brown's method in §4 extends the single-function argument by applying the fractional moment estimates directly to the product of the L-functions via the approximate functional equation for each factor. Because the method is constructed to avoid zero-distribution data for each individual L-function of degree less than 3, the cross terms in the expanded multi-function resonance product are estimated using the same zero-density-free bounds as in the single-function case. No new zero-free-region dependencies arise. We will add an explicit line-by-line comparison with the single-function Heath-Brown argument in the revised manuscript. revision: yes

  2. Referee: [§5.3] §5.3 (Joint resonance on the critical line): the claimed improvement over Heap-Li is stated to hold for any number of functions, yet the moment bound in the multi-L-function case appears to reduce the effective range of the resonance parameter by a factor depending on the number of functions; it is not shown that this reduction remains harmless for the lower bound on the maximum when the number of L-functions grows.

    Authors: The moment bound in the multi-function setting does reduce the admissible range of the resonance parameter by a multiplicative factor depending on the number k of L-functions. However, this factor is independent of the height T and only influences the implied constant in the lower bound for the joint maximum. Since the result asserts that for every fixed k the joint maximum attains the expected large value (with a constant depending on k), the reduction does not invalidate the asymptotic lower bound. When k increases the constant deteriorates, but the statement remains valid for each fixed k. We will add a clarifying remark on the k-dependence in §5.3. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external methods and a newly introduced variation

full rationale

The paper's central unconditional claim for simultaneous large values of arbitrarily many GL(1) and GL(2) L-functions on the critical line is obtained by combining Soundararajan's resonance method with a variation of Heath-Brown's method for fractional moments, which the paper explicitly introduces to avoid zero-distribution information for degree <3 L-functions. This variation is presented as original work rather than derived from prior self-citations or fitted parameters. Off the line, results assume external zero-density estimates. No steps in the provided abstract or description reduce by construction to self-definitional inputs, renamed known results, or load-bearing self-citations whose authors overlap with the present work. The improvement over Heap-Li is therefore not forced by internal redefinition or self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of L-functions and assumes zero-density estimates for the off-critical-line portion.

axioms (2)
  • standard math Standard analytic properties of primitive GL(1) and GL(2) L-functions including functional equations and Euler products.
    Invoked as background throughout the resonance method application.
  • domain assumption Zero-density estimates for the joint distribution results to the right of the critical line.
    Explicitly required for the off-critical-line study and partial recovery of prior results.

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