Large values of $L(σ,χ)$ for subgroups of characters

Abstract: We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,χ)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $χ$ 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.

• The paper presents precise unconditional and conditional (GRH-based) lower bounds for |L(σ,χ)| in thin subgroups of Dirichlet characters.
• It employs a refined resonance method with optimized truncated Euler products to achieve sharp estimates and new zero density bounds.
• The work applies these findings to improve understanding of primitive root distributions and advance analytic techniques in number theory.

On Large Values of $L(\sigma,\chi)$ in Thin Subgroups of Characters

Introduction and Motivation

The paper "Large values of $L(\sigma,\chi)$ for subgroups of characters" (2604.02960) investigates the extreme value distribution of Dirichlet $L$-functions $L(s,\chi)$ when the character $\chi$ ranges over thin subgroups of the full group of Dirichlet characters modulo $q$. While there is a robust literature on the value-distribution and large value behavior of $L$-functions when $\chi$ varies over all characters (or over natural coset or fixed order subsets), comparatively little was known about the regime of extremely thin subgroups. This work fills that gap by developing both unconditional and conditional (GRH-based) lower bounds for the maximum of $|L(\sigma,\chi)|$ as $\chi$ varies over a subgroup, as well as new zero density results and a sharpened mean-value bound for corresponding character sums. These findings have further implications for classical analytic number theory problems, including the fine-scale distribution of powers of primitive roots modulo a prime.

Main Results on Large Values

Lower Bounds for $L(\sigma,\chi)$0 in Thin Subgroups

The authors establish the existence of extremely large values of $L(\sigma,\chi)$1, even when $L(\sigma,\chi)$2 is restricted to a proper, "thin" subgroup $L(\sigma,\chi)$3, provided the order $L(\sigma,\chi)$4 is at least $L(\sigma,\chi)$5 (unconditionally) or $L(\sigma,\chi)$6 (conditionally on GRH). Specifically, for any $L(\sigma,\chi)$7 and sufficiently large $L(\sigma,\chi)$8, there exists a non-principal $L(\sigma,\chi)$9 with

$L$0

where $L$1 is the Euler-Mascheroni constant. The proof utilizes a refinement of the resonance method that exploits the algebraic structure of the subgroup, together with a precise approximation of $L$2 by truncated Euler products and a highly optimized parameter selection strategy.

Large Values in the Critical Strip

For $L$3, the authors prove that when the subgroup size $L$4 exceeds $L$5 (or, assuming GRH, $L$6 with $L$7), the maximum (over $L$8) of $L$9 is at least of order

$L(s,\chi)$0

as $L(s,\chi)$1. This matches the order of the conjectured correct size for the maximum value, even in these highly restricted character sets. The result is conditional on new zero density estimates (see below) and leverages a careful analysis of truncated Dirichlet series, together with refined resonance polynomials adapted to the subgroup structure.

Large Values at the Central Point $L(s,\chi)$2

At $L(s,\chi)$3, the authors show that if $L(s,\chi)$4 (for any fixed $L(s,\chi)$5), there exists a non-principal even character $L(s,\chi)$6 in an order-$L(s,\chi)$7 subgroup of $L(s,\chi)$8 for which

$L(s,\chi)$9

The proof combines the resonance method with a variant of the de la Bretèche–Tenenbaum framework and is structured to maximize the main term arising from highly structured GCD sums.

These extreme value results demonstrate the presence of large $\chi$0-values even in subgroups whose size is much smaller than the ambient character group, with the lower size threshold precisely quantified relative to $\chi$1 and $\chi$2. They are strict generalizations of all previously established large value results for full groups or cosets.

Zero Density Estimates and Mean Value Bounds

A key technical advance of the paper is the derivation of nontrivial zero density bounds for Dirichlet $\chi$3-functions averaged over thin subgroups. Given a subgroup $\chi$4 of order $\chi$5, for $\chi$6 and $\chi$7, the total number of zeros (with $\chi$8 and $\chi$9) summed over $q$0 admits the bound

$q$1

This improves on the trivial bound $q$2 as soon as $q$3, and crucially underpins the large value results for $q$4.

The proof relies on an improved mean value bound for character sums over sets with small multiplicative doubling, generalizing classical results of Montgomery. Specifically, for a set $q$5 with $q$6 and positive integer $q$7, the mean value

$q$8

is bounded by

$q$9

a result which is sharp in several regimes and has important implications for the analysis of $L$0-functions and character sums.

Applications to Distribution of Powers of Primitive Roots

As an application of these mean value estimates, the authors unconditionally improve results of Montgomery and Rudnick–Zaharescu on the distribution of differences between small powers of a primitive root modulo a prime. In particular, they show that if $L$1 and $L$2 (for any fixed $L$3), the variance in the number of times an interval of length $L$4 is hit by the powers $L$5 has the asymptotic

$L$6

as $L$7. Additionally, for $L$8, the pair correlation of the sequence of powers is Poissonian in the uniform $L$9-aspect, improving previous GRH-based and "almost all" results to fully unconditional results in a wider parameter range.

Theoretical and Practical Implications

The paper rigorously quantifies the boundary at which large value phenomena for Dirichlet $\chi$0-functions can be observed within subgroups, clarifying the interaction between the algebraic complexity (subgroup size) and analytic properties (location and density of zeros, moments of $\chi$1-functions). The zero density bounds in subgroups and the improved character sum mean values provide tools that are likely to prove useful well beyond the current setting, with potential applications to distribution questions for other $\chi$2-functions in families indexed by algebraic data (e.g., Galois orbits, torsion points, orbits of group actions).

Practically, the ability to isolate large $\chi$3-values within highly structured or restricted sets is highly relevant for applications in analytic number theory, cryptography (primitive root and character-related constructions), and the arithmetic statistics of families of automorphic forms. The improvements to the local spacing statistics for powers of primitive roots may inform further work in random matrix theory analogues for thin or non-uniform families.

Future Perspectives

This work naturally suggests further investigation into more refined sub-structural properties of character groups, particularly exploring whether even thinner subsets (such as small intervals or non-subgroup structured sets) can support similar extreme value phenomena. The techniques may also be adaptable to more general $\chi$4-functions (e.g., higher degree, function field analogues, or $\chi$5-functions of automorphic representations), and to higher moments or joint value-distribution results. The extension of unconditional results to even smaller $\chi$6 in primitive root power statistics remains an open and compelling challenge.

Conclusion

This paper establishes precise lower bounds for the maximum size of $\chi$7 over thin subgroups of Dirichlet characters, demonstrates strong unconditional zero density and mean value estimates, and applies these results to improve understanding of the fine-scale distribution of primitive root powers. The adoption of additive combinatorics and resonance methods within the analytic number theory framework advances both theory and methodology, opening new directions for the study of value-distribution and arithmetic statistics in constrained algebraic families.