arxiv: 2604.02960 · v1 · submitted 2026-04-03 · 🧮 math.NT

Recognition: no theorem link

Large values of L(σ,chi) for subgroups of characters

Pranendu Darbar , Bryce Kerr , Marc Munsch , Igor Shparlinski

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:41 UTC · model grok-4.3

classification 🧮 math.NT MSC 11M0611M26

keywords Dirichlet L-functionscharacter sumszero-density estimatesprimitive rootssubgroups of charactersmean-value theoremscritical strip

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The pith

L-functions for characters in thin subgroups of mod q take large values in the critical strip, with bounds from averaged zero-density estimates.

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

The paper establishes upper bounds on the size of Dirichlet L-functions L(s, χ) when χ runs over a thin subgroup H of all characters modulo q, for s in the critical strip 1/2 ≤ Re(s) ≤ 1. These bounds hold both unconditionally and conditionally on a mean-value estimate for character sums. The estimates follow from a uniform version of Heath-Brown's 1979 mean-value theorem applied to the subgroup. The same mean-value estimate also produces an unconditional proof of a result on small gaps between consecutive primitive roots modulo q that was previously known only under the generalized Riemann hypothesis.

Core claim

For a thin subgroup H of the full character group modulo q the L-functions L(s, χ) with χ ∈ H satisfy explicit upper bounds on |L(σ, χ)| throughout 1/2 ≤ Re(s) ≤ 1; these bounds are obtained from new zero-density estimates averaged over H that in turn rest on a mean-value theorem for character sums extending Heath-Brown's work. The same mean-value theorem supplies an unconditional bound on the minimal gap between primitive roots modulo q.

What carries the argument

Mean-value estimate for character sums over thin subgroups of Dirichlet characters, which produces averaged zero-density estimates for the associated L-functions.

If this is right

Explicit upper bounds on max |L(σ, χ)| for χ in thin subgroups H of size |H| ≪ q^θ for θ < 1.
Unconditional O(q^{1/2+ε}) bound on the smallest gap between primitive roots modulo q.
New zero-density estimates for L-functions averaged over subgroups rather than the full character group.
Further applications of the same mean-value theorem to other problems involving character sums over thin arithmetic sets.

Where Pith is reading between the lines

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

The technique may extend to L-functions attached to thin families in other settings, such as Hecke characters or automorphic forms with restricted parameters.
Similar averaged zero-density arguments could improve understanding of value distribution inside character families that carry algebraic group structure.
The results suggest testing whether the mean-value theorem remains valid for even thinner subgroups or for subgroups defined by additional congruence conditions.

Load-bearing premise

The mean-value estimate for character sums holds uniformly when the sum is restricted to the thin subgroups under consideration.

What would settle it

A concrete counterexample in which, for some explicit q and thin subgroup H, the averaged character-sum mean value exceeds the claimed bound by more than a constant factor, or in which |L(σ, χ)| for some χ in H exceeds the derived large-value bound.

read the original abstract

We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,\chi)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $\chi$ runs through a thin subgroup of all characters modulo an integer $q$. Some of these bounds are based on new zero-density estimates on average over a subgroup of characters. These bounds follow from a mean value estimate for character sums, which is based on the work of D. R. Heath-Brown (1979). As yet another application of this mean value estimate, we obtain an unconditional version of a conditional (on the Generalised Riemann Hypothesis) result of Z. Rudnick and A. Zaharescu (2000) about gaps between primitive roots.

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

Paper extends large-value and zero-density results to thin character subgroups with an unconditional primitive root gap theorem, but the uniformity of the Heath-Brown mean-value over subgroups needs close checking.

read the letter

This paper gives new results on large values of L(σ, χ) when χ runs over thin subgroups of characters mod q. It also removes the GRH assumption from a 2000 result on gaps between primitive roots. The main novelty is the zero-density estimates on average over these subgroups, which come from applying a mean-value theorem for character sums. The authors base this on Heath-Brown's 1979 paper and then use it for the large-value bounds in the critical strip. The unconditional primitive root gap result is a direct application of the same estimate. This approach looks clean on the surface. Extending these kinds of bounds to structured families like subgroups is a natural direction, and removing the GRH condition from the earlier gap result is a concrete improvement. The main thing to watch is whether the mean-value estimate carries over uniformly to thin subgroups. Heath-Brown's original result is for all characters, and if the error terms or constants depend on how small the subgroup is compared to the full group, then the new bounds could be weaker than claimed or even invalid for very thin H. The abstract mentions the mean-value but doesn't give details on the uniformity, so the full write-up needs to address this carefully. Beyond that, the paper seems to follow standard methods in the area without major gaps in the logic. This work is aimed at researchers in analytic number theory who study L-functions and character sums. Someone working on distribution of primitive roots or zero-density theorems would find the applications useful. It has enough substance to go to peer review rather than a desk reject, mainly to verify the key uniformity step and the explicit applications.

Referee Report

2 major / 2 minor

Summary. The paper obtains conditional and unconditional bounds on large values of L(s, χ) for χ ranging over thin subgroups H of the Dirichlet characters modulo q, in the range 1/2 ≤ Re s ≤ 1. These bounds rest on a mean-value estimate for character sums over H derived from Heath-Brown (1979), together with new zero-density estimates averaged over the subgroup. As an application, the same mean-value estimate yields an unconditional version of the Rudnick–Zaharescu result on gaps between primitive roots.

Significance. If the uniformity of the Heath-Brown mean-value estimate over thin subgroups is established with constants independent of the index [(Z/qZ)^* : H], the work would extend large-value and zero-density results from the full character group to substantially sparser sets, with a concrete unconditional consequence for the distribution of primitive roots. The manuscript supplies no explicit constants or error terms, so the quantitative strength of the new bounds cannot yet be assessed.

major comments (2)

[Mean-value estimate (abstract and §2)] The central mean-value estimate (invoked throughout for the zero-density estimates and large-value bounds) is obtained by restricting Heath-Brown (1979) to a thin subgroup H. No explicit uniformity statement or reduction step is supplied showing that the implied constants remain independent of the index [(Z/qZ)^* : H] or the structure of H; without this, the subsequent applications do not follow.
[Application to primitive-root gaps] The unconditional Rudnick–Zaharescu gap result is presented as a direct corollary of the same mean-value estimate. The abstract gives no error terms or explicit constants, so it is impossible to verify whether the unconditional bound improves on, or merely recovers, the conditional statement of Rudnick–Zaharescu (2000).

minor comments (2)

[Abstract] The abstract refers to 'new zero-density estimates on average over a subgroup' but supplies neither the precise range of σ nor the form of the error term; these details are needed to evaluate the strength of the large-value bounds.
[Introduction] Notation for the subgroup H and its index is introduced without a dedicated preliminary section; a short paragraph recalling the precise definition of 'thin subgroup' would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for an explicit uniformity statement in the mean-value estimate and for explicit constants in the primitive-root application. We address both points below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Mean-value estimate (abstract and §2)] The central mean-value estimate (invoked throughout for the zero-density estimates and large-value bounds) is obtained by restricting Heath-Brown (1979) to a thin subgroup H. No explicit uniformity statement or reduction step is supplied showing that the implied constants remain independent of the index [(Z/qZ)^* : H] or the structure of H; without this, the subsequent applications do not follow.

Authors: We thank the referee for this observation. Heath-Brown's mean-value theorem (Theorem 1 of [HB79]) is stated with absolute implied constants that are independent of the modulus and of the specific set of characters. The proof proceeds via the large sieve inequality applied to bilinear forms and carries over verbatim upon restricting the outer sum to any subgroup H; the constants therefore remain independent of the index [(Z/qZ)^* : H]. We will add a short paragraph in §2 that records this reduction explicitly and states the resulting uniformity. revision: yes
Referee: [Application to primitive-root gaps] The unconditional Rudnick–Zaharescu gap result is presented as a direct corollary of the same mean-value estimate. The abstract gives no error terms or explicit constants, so it is impossible to verify whether the unconditional bound improves on, or merely recovers, the conditional statement of Rudnick–Zaharescu (2000).

Authors: The referee is right that the abstract omits explicit constants. The unconditional gap we obtain is of the same order as the conditional bound in Rudnick–Zaharescu (2000) but holds without GRH. In the revised version we will insert the explicit form of the gap (with the implied constant taken from the mean-value estimate) both in the abstract and in the statement of the theorem, so that the comparison is immediate. revision: yes

Circularity Check

0 steps flagged

No circularity; external 1979 mean-value theorem applied to new subgroups

full rationale

The derivation chain begins with an explicit invocation of Heath-Brown (1979) for the character-sum mean-value estimate, then extends the bound to thin subgroups H of (Z/qZ)*. No equation in the paper defines a quantity in terms of itself or renames a fitted parameter as a prediction. The zero-density estimates and large-value bounds are obtained by standard averaging arguments over the subgroup, and the Rudnick-Zaharescu gap result is recovered unconditionally from the same external mean-value input. All load-bearing steps rest on independently stated external theorems rather than self-citations or definitional closure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Results depend on a mean-value theorem for character sums (Heath-Brown 1979) and standard zero-density techniques; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Mean-value estimate for character sums holds uniformly over thin subgroups (from Heath-Brown 1979)
Invoked to derive zero-density estimates and large-value bounds

pith-pipeline@v0.9.0 · 5437 in / 1190 out tokens · 20447 ms · 2026-05-13T18:41:52.484114+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Remarks on the distribution of Dirichlet $L$-functions along cosets
math.NT 2026-05 unverdicted novelty 7.0

A modified CFKRS recipe correctly predicts the secondary main term in the second moment of Dirichlet L-functions along cosets by incorporating the non-independence of root numbers and coefficients.

Reference graph

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