Negative discrete second moments of Dirichlet $L$-functions

Technical Summary: Negative Discrete Second Moments of Dirichlet L-Functions

Problem Statement
This paper investigates lower bounds for two specific discrete second moments of Dirichlet $L$-functions $L(s, \chi)$, taken over their non-trivial zeros $\rho = \frac{1}{2} + i\gamma$. The study focuses on the uniformity of these bounds with respect to the conductor $q$, particularly in regimes where $q$ grows with the height $T$ of the zeros. The two moments analyzed are:

1. The Reciprocal Moment: $\sum_{0<\gamma \le T} |L'(\rho, \chi)|^{-2}$.
2. The Ratio Moment: $\sum_{0<\gamma \le T} \left| \frac{L(2\rho, \chi^2)}{L'(\rho, \chi)} \right|^2$.

These quantities generalize conjectures and theorems previously established for the Riemann zeta-function ($q=1$) by Gonek, Milinovich, Ng, and Sinha. The central challenge addressed is determining how the conductor $q$ affects the magnitude of these sums, specifically identifying the "conductor deficit" that arises when $q$ is large relative to $T$.

Methodology
The author employs the mollification technique, adapting the strategy of Milinovich and Ng (which itself relies on the work of Rudnick and Soundararajan). The core approach involves:

• Mollifiers: Two distinct Dirichlet polynomials are constructed to approximate the reciprocals of the terms in the moments.
  • For the reciprocal moment, a Möbius-twisted polynomial $L_\psi(s) = \sum_{n \le \xi} \mu(n)\psi(n)P_n n^{-s}$ is used, corresponding to the Dirichlet series for $1/L(s, \chi)$.
  • For the ratio moment, a Liouville-twisted polynomial $\tilde{L}_\psi(s) = \sum_{n \le \xi} \lambda(n)\psi(n)P_n n^{-s}$ is used, corresponding to the series for $L(2s, \chi^2)/L(s, \chi)$.
  • Here, $\xi = T^\vartheta$ is the mollifier length, and $P(x)$ is a polynomial with $P(0)=0, P(1)=1$.
• Cauchy-Schwarz Inequality: The moments are bounded below by applying the Cauchy-Schwarz inequality to sums of the form $\sum \frac{1}{L'(\rho, \chi)} L_\chi(\rho)$ (and the ratio analogue). This reduces the problem to estimating the first moment ($M_1$) and the second moment ($M_2$) of the mollified functions over the zeros.
• Contour Integration: The sums $M_1$ and $M_2$ are evaluated using the residue theorem over rectangular contours. The integrals along the vertical edges yield the main terms, while horizontal edges are bounded using size estimates for $L$-functions.
• Uniformity: A critical technical component is maintaining uniformity in the conductor $q$. This requires careful handling of the arithmetic factors (involving $\phi(q)$ and $A_q$) and the error terms in the mean value theorems for Dirichlet polynomials. The Generalised Riemann Hypothesis (GRH) and the simplicity of zeros are assumed throughout.

Key Results
The paper establishes lower bounds for both moments that depend on the ratio $\beta = \frac{\log T}{\log qT}$.

• Theorem 5 (Reciprocal Moment): Assuming GRH and simple zeros, for $3 \le q \le T^A$, the sum is bounded below by:
  $$(1-\varepsilon) \frac{A_q}{2\pi} \frac{\beta}{1+\beta} T$$
  where $A_q = \frac{6}{\pi^2} \prod_{p|q} \frac{p}{p+1}$.

• Theorem 6 (Ratio Moment): Assuming GRH for $L(s, \chi)$ (and simple zeros), for $3 \le q \le T^{1-\delta}$, the sum is bounded below by:
  $$(1-\varepsilon) \frac{\phi(q)}{q} \frac{1}{2\pi} \frac{\beta}{1+\beta} T$$
  The restriction $q \le T^{1-\delta}$ arises because the proof does not assume GRH for the character $\chi^2$ appearing in the numerator; assuming GRH for $\chi^2$ would extend the range to $q \le T^A$.

• Asymptotic Behavior:

  • Small Conductors: If $\log q = o(\log T)$ (e.g., fixed $q$), then $\beta \to 1$. The bounds recover half of the conjectured asymptotic size (matching the $q=1$ results of Milinovich-Ng and Sinha).
  • Power Conductors: If $q = T^A$, then $\beta = \frac{1}{1+A}$, and the captured proportion becomes $\frac{1}{2+A}$. For example, at $A=1$, the bound is $1/3$ of the conjectured size.

• Higher Moments: By applying Hölder's inequality, the author derives lower bounds for higher moments ($2k$) in Corollaries 11 and 12. These bounds are of order $T/(\log qT)^{k-1}$, which are significantly smaller than the conjectured sizes for $k>1$, but the paper emphasizes their uniformity in $q$ rather than their sharpness.

Conjectures and Significance
The paper posits that the derived lower bounds capture exactly the proportion $\frac{\beta}{1+\beta}$ of the true asymptotic values.

• Conjectures 7 and 8: The author conjectures that the true asymptotics are $\frac{A_q}{2\pi}T$ and $\frac{\phi(q)}{q}\frac{T}{2\pi}$ respectively, uniformly for $q \le T^A$.
• Source of the Deficit: The paper argues that the factor $\frac{\beta}{1+\beta}$ is not a numerical accident but a structural feature of the diagonal Cauchy-Schwarz construction. The mean value theorem for Dirichlet polynomials limits the mollifier length $\xi$ to $T^{1-\varepsilon}$, independent of $q$. However, the zero density grows with $\log qT$. The "deficit" is the ratio of the value of the function $u/(1+u)$ at the cap $u=\beta$ to its limit as $u \to \infty$.
• Family Averages: The author suggests that averaging over the family of primitive characters (Conjectures 9 and 10) might allow for longer mollifiers (up to $(qT)^{1-\varepsilon}$), potentially improving the captured proportion from $\frac{1}{2+A}$ to $\frac{1}{2}$ in the power-conductor regime. However, the paper does not prove this, noting that it would require new uniform explicit formulae.

Conclusion
The paper provides the first uniform lower bounds for negative discrete moments of Dirichlet $L$-functions across a wide range of conductors. It successfully generalizes known $q=1$ results and quantifies the degradation of these bounds as the conductor grows. The work identifies a specific structural limitation in the current mollification method regarding the conductor aspect and formulates precise conjectures for the true asymptotic behavior, inviting further research into family-averaged techniques to overcome the current "conductor deficit."