Applications of congruence-counting bounds to Dirichlet L-functions

Ascertain whether existing upper bounds for the number of solutions to the congruence m ≡ γ n (mod q) with integers 1 ≤ m,n ≤ N and γ ranging over a subgroup Γ ⊂ Z_q^* (such as those established by Bourgain–Konyagin–Shparlinski, Cilleruelo–Garaev, and Shparlinski) can be leveraged to obtain interesting applications to Dirichlet L-functions, for example by yielding nontrivial estimates for the mean value M(𝒜, N) = (1/|𝒜|) ∑_{χ∈𝒜} |∑_{n=1}^N χ(n)| when 𝒜 is a set of characters with small multiplicative doubling, particularly in the regime of very small subgroups Γ.

Background

In the subsection on mean values of character sums over sets of characters with small multiplicative doubling, the authors introduce M(𝒜, N) and prove a new bound (Theorem on M(α, 𝒜, N)) that improves Montgomery’s estimate using Heath-Brown’s large values estimate and the Green–Ruzsa covering lemma.

They note an alternative approach: by Cauchy–Schwarz and orthogonality, estimating M(𝒜, N) can be reduced to counting solutions of m ≡ γ n (mod q) with γ in a subgroup Γ ⊂ Z_q^*. Upper bounds for such congruences are known (Bourgain–Konyagin–Shparlinski; Cilleruelo–Garaev; Shparlinski).

However, the authors explicitly state uncertainty about whether these congruence-counting results lead to interesting applications to Dirichlet L-functions, especially for very small subgroups, thus formulating an open direction.