Extreme central values of quadratic Dirichlet $L$-functions with prime conductors

Technical Summary: Extreme Central Values of Quadratic Dirichlet L-Functions with Prime Conductors

Problem Statement
The paper investigates the magnitude of the central values $L(1/2, \chi_p)$ for quadratic Dirichlet L-functions where the conductor $p$ is a prime number satisfying $p \equiv 1 \pmod 8$. While the distribution of these values is a well-studied topic (with asymptotic formulas established for the first, second, and third moments by Jutila, Goldfeld–Hoffstein, Young, and Soundararajan, among others), the specific question of how large these values can become (extreme values) remains a critical area of inquiry. The authors aim to establish a rigorous lower bound for the maximum value of $L(1/2, \chi_p)$ within a short interval of primes.

Methodology
The proof relies on Soundararajan's resonance method, a technique designed to detect large values of L-functions by constructing a "resonator" function that amplifies the central values. The core strategy involves analyzing the twisted first moment:
$$M_2(R, X) = \sum_{p \equiv 1 (8)} (\log p) L(1/2, \chi_p) R(p)^2 \Phi(p/X)$$
where $R(p)$ is a Dirichlet polynomial (the resonator) and $\Phi$ is a smooth weight function.

The authors generalize the approach taken by Baluyot and Pratt [1]. Their specific methodological innovations include:

1. Generalization of Twists: They extend the twisted first moment analysis to cases involving short Dirichlet series with large coefficients. Unlike previous applications where coefficients were bounded or satisfied strict non-vanishing conditions akin to the generalized Ramanujan conjecture, this paper allows for coefficients that grow rapidly.
2. Diagonal vs. Off-Diagonal Analysis: The proof hinges on showing that the "diagonal" terms (where the product of resonator indices is a square) dominate the "off-diagonal" terms (where the product is not a square) within specific parameter ranges.
   • Diagonal Terms: These are estimated using the prime number theorem in arithmetic progressions and properties of the function $V(y)$ derived from the approximate functional equation.
   • Off-Diagonal Terms: These are bounded by splitting the summation range into three regimes based on the size of the product of the resonator indices ($m_0r$):
     • Regime I (Small): Handled using Page's Theorem to control contributions from potential Siegel zeros (exceptional real zeros of L-functions).
     • Regime II (Medium): Controlled using zero-density estimates (specifically Montgomery's estimate) to bound the number of zeros in the critical strip.
     • Regime III (Large): Bounded using Vaughan's identity to decompose the von Mangoldt function and manage the summation over large indices.

Key Contributions and Results
The primary result is Theorem 1.1, which establishes a lower bound for the maximum central value. For sufficiently large $X$, the authors prove:
$$\max_{\substack{X < p \le 2X \\ p \equiv 1 (8)}} L(1/2, \chi_p) \ge \exp\left( \left(\sqrt{\frac{8}{45}} + o(1)\right) \frac{\sqrt{\log X}}{\sqrt{\log \log X}} \right)$$
where the constant in the $o(1)$ term is effective.

The paper achieves this by:

• Demonstrating that Baluyot–Pratt's strategy can be successfully extended to a wider class of twists, specifically those with large Dirichlet coefficients.
• Showing that when diagonal terms exhibit rapid growth, they can dominate off-diagonal contributions, a condition that allows for the derivation of the lower bound without requiring the stronger assumptions often needed for asymptotic formulas.
• Providing a detailed estimation of the error terms arising from the three regimes, ensuring the off-diagonal contribution is negligible compared to the diagonal contribution for the chosen parameter $\theta = 2/45 - 3\varepsilon$.

Significance and Claims
The authors position this work as an extension of existing resonance methods to a more challenging class of coefficients.

• Comparison to Previous Work: The paper notes that while Soundararajan's method under the Generalized Riemann Hypothesis (GRH) typically yields bounds of the form $\exp(c \sqrt{\log X / \log \log X})$ with $c=1$, the unconditional result here yields a constant $c = \sqrt{8/45} \approx 0.42$. The authors acknowledge that obtaining the conjectured optimal constant ($c=1$) or stronger bounds involving triple-logarithmic terms (as seen in recent work by Darbar and Maiti [9, 10] for $\zeta(s)$) remains an open and difficult problem.
• Methodological Scope: The paper explicitly states that it provides a lower bound rather than an asymptotic formula. As noted in Remark 1.2, obtaining a minimum result (or a precise asymptotic) for the twisted first moment would be significantly more difficult.
• Context: The work fits into the broader context of understanding the distribution of central L-values, complementing results on moments (Keating–Snaith conjectures) and extreme values for other families (Riemann zeta, automorphic L-functions).

The paper concludes by situating these results within the active field of automorphic L-functions in quadratic twist families, referencing recent developments in moments and extreme values for elliptic curves and modular forms, but does not propose new applications or future experimental directions beyond the mathematical bounds established.