Bounds for moments of Dirichlet $L$-functions to a fixed modulus

Abstract

We study the 2k-th moment of central values of the family of Dirichlet L-functions to a fixed prime modulus. We establish sharp lower bounds for all real k ≥ 0 and sharp upper bounds for k in the range 0 ≤ k ≤ 1.

Moments of Dirichlet $L$-functions to a fixed modulus over function fields

“…Our proofs of the above propositions are similar to those for Propositions 3.3-3.5 in [13]. We shall therefore omit the proof of Proposition 3.4 and be brief on the proofs of Propositions 3.2 and 3.3.…”

“…A modification of a method of M. Radziwi l l and K. Soundararajan [25] can be applied to establish sharp lower bounds for all k ≥ 1. Using a lower bound principle developed by W. Heap and K. Soundararajan [16], P. Gao [13] obtained sharp lower bounds for all k ≥ 0.…”

Bounds for moments of cubic and quartic Dirichlet $L$-functions

“…Proof. The proof is also similar to that of [8,Lemma 3.1]. We first use Hölder's inequality to bound the left side of (6.…”

“…χ)M(χ, k − 1)| 2 (1−k)/2 * χ,q |M(χ, k)| 2/k |M(χ, k − 1)| 2 k/2 . (6.2)As in the proof of[8, Lemma 3.1], we note for |z| ≤ aK/10 with 0 < a ≤ 1,We apply (6.3) with z = kP ′ i (χ), K = e 2 kα −3/4 i and a = k to see that when|P ′ i (χ)| ≤ ⌈e 2 kα −3/4 i ⌉/10, M i (χ, k) = exp(kP ′ i (χ))(1 + O   exp(k|P ′ i (χ)|) kP ′ i (χ)) 1 + O ke −e 2 kα −3/4 i Similarly, we have M i (χ, k − 1) = exp ((k − 1)P ′ i (χ)) 1 + O e −e 2 kα −3/4 iThe above estimations then yield that if|P ′ i (χ)| ≤ ⌈e 2 kα −3/4 i ⌉/10, then |M i (χ, k) M i (χ, k − 1)| 2 = exp(2kℜP ′ i (χ)) 1 + O e −e 2 kα −3/=|M j (χ, k)| 2 1 + O e −e 2 kα −3/4 iOn the other hand, we notice that when|P ′ i (χ) ≥ ⌈e 2 kα −3/4 i ⌉/10, |M i (χ, k)| ≤ ⌈e 2 kα −3/4 i ⌉ r=0 |P ′ i (χ)| r r! ≤ |P ′ i (χ)| ⌈e 2 kα −3/4Observe that the same bound above also holds for |M i (χ, k − 1)|.…”

“…We omit the proof of the above lemma since its proof is similar to that of [8,Lemma 3.1]. It follows from this lemma that in order to establish Theorem 1.2, it suffices to prove the following two propositions.…”