Extreme values of derivatives of Dirichlet $L$-functions

Technical Summary: Extreme Values of Derivatives of Dirichlet L-Functions

Problem Statement
The paper addresses the problem of establishing lower bounds for the extreme values of the $\ell$-th derivatives of Dirichlet $L$-functions, denoted as $L^{(\ell)}(s, \chi)$, within the critical strip range $1/2 < \sigma < 1$. While the extreme values of the $L$-functions themselves ($\ell=0$) have been extensively studied—most notably by Aistleitner, Mahatab, Munsch, and Peyrot [3] who established sharp lower bounds for $|L(1, \chi)|$ and $|L(\sigma, \chi)|$—the behavior of their derivatives in the range $1/2 < \sigma < 1$ remained less explored. Previous work by D. Yang [14] had successfully analyzed derivatives at $s=1$, but the extension to the open interval $(1/2, 1)$ required new techniques.

Methodology
The authors employ the long resonator method, a refinement of the resonance technique originally introduced by Voronin [11] and improved by Soundararajan [10] and others. The core strategy involves constructing a specific "resonator" function $R(\chi)$ designed to amplify the values of $L^{(\ell)}(\sigma, \chi)$ for a specific non-principal character $\chi$.

The technical execution proceeds as follows:

1. Truncation of the Series: The authors first establish that for sufficiently large primes $q$, the derivative $L^{(\ell)}(\sigma, \chi)$ can be effectively approximated by a truncated Dirichlet series $P(\chi) = \sum_{n \le q} (\log n)^\ell \chi(n) n^{-\sigma}$. The error term is shown to be negligible ($O(q^{1/2-\sigma}(\log q)^{\ell+1})$) using the Pólya-Vinogradov inequality and partial summation (Lemma 2.1).
2. Resonator Construction: A completely multiplicative function $r(n)$ is defined such that $r(p) = 1/2$ for primes $p \le Y$ and $0$ otherwise, where $Y = \eta \log q \log \log q$. The resonator is defined as $R(\chi) = \sum r(n)\chi(n)$.
3. Moment Analysis: The proof relies on analyzing the ratio of two sums over the set of Dirichlet characters modulo $q$:
   • $S_1 = \sum |R(\chi)|^2$
   • $S_2 = \sum P(\chi)|R(\chi)|^2$
     By expanding these sums and utilizing the orthogonality of Dirichlet characters, the authors derive a lower bound for the maximum value of $|P(\chi)|$ via the inequality $\max |P(\chi)| \ge S_2/S_1$.
4. Asymptotic Estimation: The authors carefully estimate the sums $S_1$ and $S_2$, isolating the contribution of the principal character (which is shown to be negligible) and focusing on non-principal characters. The estimation involves Euler products and the Prime Number Theorem to evaluate the growth of the resonator's magnitude.

Key Contributions and Results
The primary contribution of the paper is Theorem 1.1, which establishes that for any fixed integer $\ell \ge 0$ and $\sigma \in (1/2, 1)$, there exists a constant $c_{\sigma, \ell} > 0$ such that for every sufficiently large prime $q$, there exists a non-principal character $\chi \pmod q$ satisfying:
$$|L^{(\ell)}(\sigma, \chi)| \ge \exp\left( c_{\sigma, \ell} \frac{(\log q)^{1-\sigma}}{(\log \log q)^\sigma} \right)$$

This result demonstrates that the derivatives of Dirichlet $L$-functions can attain extreme values of the same order of magnitude as the original $L$-functions in the range $1/2 < \sigma < 1$. Specifically:

• In the case $\ell=0$, the theorem recovers the known result by Aistleitner et al. [3] regarding $|L(\sigma, \chi)|$.
• The work generalizes previous findings on the Riemann zeta function (specifically [12, Theorem 2(B)]) to the context of Dirichlet $L$-functions.
• It extends the scope of D. Yang's work [14] from the boundary point $s=1$ to the interior of the critical strip.

Significance
The paper claims significance by bridging a gap in the understanding of the extreme value distribution of $L$-function derivatives. By proving that derivatives do not decay significantly in magnitude compared to the functions themselves within the critical strip, the authors reinforce the connection between the analytic properties of these derivatives and fundamental problems in number theory, such as character sum estimates and the distribution of quadratic residues. The successful application of the long resonator method to this specific problem highlights the method's versatility in handling higher-order derivatives in the critical strip.