Joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives

1. Problem Statement

The paper addresses the problem of establishing joint extreme values for Dirichlet $L$-functions and their logarithmic derivatives. Specifically, the author investigates the simultaneous behavior of the values of $L(s, \chi^j)$ and $L'/L(s, \chi^j)$ for $j = 1, 2, \dots, \ell$, where $\chi$ is a Dirichlet character modulo a large prime $q$, and $\ell$ is a fixed positive integer.

The study covers two distinct regions of the complex plane:

1. The point $s=1$: Investigating the product $\prod_{j=1}^\ell |L(1, \chi^j)|$ and the real part of the product of logarithmic derivatives.
2. The critical strip $\sigma \in (1/2, 1)$: Investigating the product $\prod_{j=1}^\ell |L(\sigma, \chi^j)|$ and the corresponding logarithmic derivatives.

The core question is whether there exist characters $\chi$ such that these powers simultaneously attain large values, and if so, what the precise asymptotic lower bounds are for these joint extreme values. This extends previous work on single $L$-functions (Aistleitner et al., Granville & Soundararajan) and joint values of the Riemann zeta function at harmonic points (Yang & Zhao).

2. Methodology

The primary tool employed is the Resonance Method, specifically the Long Resonator Method introduced by Aistleitner and refined by Bondarenko and Seip. The methodology involves the following steps:

• Approximation by Dirichlet Polynomials:
  The $L$-functions and their derivatives are approximated by finite Dirichlet polynomials (truncated Euler products or sums involving the von Mangoldt function $\Lambda(n)$).

  • For $s=1$, $L(1, \chi)$ is approximated by $L(1, \chi; Y) = \prod_{p \le Y} (1 - \chi(p)p^{-1})^{-1}$.
  • For $s=\sigma$, the logarithm $\log L(\sigma, \chi)$ is approximated by $\sum_{n \le Y} \Lambda(n)\chi(n)n^{-\sigma} \log n$.
  • For logarithmic derivatives, $-L'/L(s, \chi)$ is approximated by $\sum_{n \le Y} \Lambda(n)\chi(n)n^{-s}$.
  • These approximations rely on zero-density estimates (Lemma 2.3) to ensure that the set of "exceptional" characters where the approximation fails is negligible compared to the total number of characters.

• Construction of the Resonator:
  A resonator function $R(\chi)$ is constructed to amplify the values of the target $L$-functions for specific characters. It is defined as:
  $$R(\chi) = \prod_{p \le X} (1 - r(p)\chi(p))^{-1} = \sum_{n} r(n)\chi(n)$$
  where $r(n)$ is a completely multiplicative function. The coefficients $r(p)$ are chosen to be close to 1 for small primes (e.g., $r(p) = 1 - p/X$ or $1 - (p/X)^\sigma$) to maximize the correlation between the resonator and the $L$-function values.

• Moment Calculations (The Ratio Method):
  The proof relies on estimating the ratio of two sums over the group of characters $G_q$:
  $$\frac{S_2}{S_1} = \frac{\sum_{\chi} \left( \prod_{j=1}^\ell \text{Target}_j(\chi) \right) |R(\chi)|^2}{\sum_{\chi} |R(\chi)|^2}$$
  By the pigeonhole principle, there must exist at least one character $\chi$ such that the target value is at least $S_2/S_1$.

  • $S_1$ is the second moment of the resonator.
  • $S_2$ involves the product of the target functions and the resonator.
  • Orthogonality of characters is used to evaluate these sums, reducing them to sums over integers satisfying congruence conditions modulo $q$. Since $q$ is prime, these conditions simplify significantly (e.g., $Km \equiv n \pmod q \implies m \equiv n \pmod q$ if $q \nmid K$).

• Asymptotic Analysis:
  The resulting sums are evaluated using the Prime Number Theorem and Mertens' theorems. The author carefully analyzes the interaction terms between different powers $j$ to derive the secondary terms in the asymptotic expansions.

3. Key Contributions and Results

The paper establishes four main theorems providing explicit lower bounds for joint extreme values.

Theorem 1.1: Joint Extreme Values at $s=1$

For a fixed integer $\ell \ge 1$ and sufficiently large prime $q$, there exists a character $\chi$ with order $>\ell$ such that:
$$\prod_{j=1}^\ell |L(1, \chi^j)| \ge e^{\ell\gamma} \left\{ (\log_2 q)^\ell + \ell(\log_3 q - C(\ell))(\log_2 q)^{\ell-1} + O((\log_2 q)^{\ell-2}(\log_3 q)^2) \right\}$$
where $C(\ell) = \frac{\ell+1}{2} + \log_2 4$.

• Significance: This recovers the result of Aistleitner et al. for $\ell=1$ but provides a sharper error term and explicitly quantifies the "interaction" between different powers, showing that extreme values are slightly compressed as $\ell$ increases.

Theorem 1.2: Joint Extreme Values in the Critical Strip ($1/2 < \sigma < 1$)

There exists a character $\chi$ such that:
$$\prod_{j=1}^\ell |L(\sigma, \chi^j)| \ge \exp \left\{ (\kappa^{1-\sigma} S(\sigma, \ell) + o(1)) \frac{(\log q)^{1-\sigma}}{(\log_2 q)^\sigma} \right\}$$
where $S(\sigma, \ell)$ is an explicit constant involving an alternating binomial sum.

• Significance: The constant $S(\sigma, \ell)$ captures the complex interplay of the powers. The growth of the bound is suppressed when $\sigma$ is close to $1/2$ due to cancellation effects in the sum, but increases significantly as $\sigma \to 1$.

Theorem 1.3: Joint Extreme Values of Logarithmic Derivatives at $s=1$

There exists a character $\chi$ such that:
$$(-1)^\ell \text{Re} \prod_{j=1}^\ell \frac{L'}{L}(1, \chi^j) \ge (\log_2 q)^\ell + (\ell \log_3 q + Q(\ell))(\log_2 q)^{\ell-1} + O(\dots)$$
where $Q(\ell)$ is an explicit negative constant decreasing rapidly with $\ell$.

• Significance: This improves upon D. Yang's previous work by providing a more explicit secondary term and a sharper error term. It confirms that the logarithmic derivatives also exhibit strong joint extreme behavior.

Theorem 1.4: Joint Extreme Values of Logarithmic Derivatives in the Critical Strip

For $1 \le \ell < (2-2\sigma)^{-1}$, there exists a character $\chi$ such that:
$$(-1)^\ell \text{Re} \prod_{j=1}^\ell \frac{L'}{L}(\sigma, \chi^j) \ge (\eta^{\ell(1-\sigma)} H(\sigma, \ell) + o(1)) (\log q)^{\ell(1-\sigma)} (\log_2 q)^{\ell(1-\sigma)}$$
where $H(\sigma, \ell)$ is an explicit constant involving factorials and products of $\sigma$.

• Significance: This extends the study of logarithmic derivatives to the critical strip. The condition on $\ell$ (bounded by $(2-2\sigma)^{-1}$) reflects the technical constraints of the resonance method in this region.

4. Significance and Implications

1. Structural Correlation: The results demonstrate that Dirichlet characters attaining extreme values are not independent. Instead, the powers $\chi, \chi^2, \dots, \chi^\ell$ tend to attain extreme values simultaneously. This reveals a strong correlation within the multiplicative group of characters, suggesting that the extreme values are driven by a common underlying multiplicative structure.
2. Refinement of Existing Bounds: The paper provides sharper error terms and more precise secondary terms compared to previous literature (Aistleitner et al., Yang). This allows for a more accurate understanding of the distribution of $L$-values.
3. Connection to Zero Distribution: Since logarithmic derivatives are closely related to the distribution of zeros of $L$-functions, the joint extreme values of $L'/L$ suggest that for specific characters, multiple functions $L(s, \chi^j)$ may simultaneously approach zeros near the same point.
4. Methodological Advancement: The paper successfully adapts the resonance method to handle products of $L$-functions and their derivatives across different powers, overcoming the combinatorial complexity introduced by the interaction of multiple indices $j$.

In summary, Shengbo Zhao's work significantly advances the understanding of the extreme value distribution of Dirichlet $L$-functions by proving that their powers and logarithmic derivatives exhibit synchronized extreme behavior, with precise quantitative bounds derived via an optimized resonance method.