A note on a conjecture of Ng

Technical Summary of "A Note on a Conjecture of Ng"

Problem Statement
This note addresses Question 5, Problem 1 from the Comparative Prime Number Theory Problem List, originally suggested by Nathan Ng. The central problem concerns the asymptotic behavior of the second moment of a specific ratio of zeta functions summed over the non-trivial zeros of the Riemann zeta function, $\zeta(s)$.

Ng's Conjecture 1 posits that, assuming the Riemann Hypothesis (RH) and the simplicity of the non-trivial zeros $\rho = 1/2 + i\gamma$, the following asymptotic holds as $T \to \infty$:
$$\sum_{0<\gamma \le T} \left| \frac{\zeta(2\rho)}{\zeta'(\rho)} \right|^2 \sim \frac{T}{2\pi}.$$
The author notes that while Ng's original statement implicitly assumed RH, he has indicated in private communication that the conjecture likely holds assuming only the simplicity of the zeros. The current work proceeds under both assumptions to ensure the summands are well-defined and to utilize specific symmetries in the proof.

Methodology
The proof strategy follows the approach of Milinovich and Ng, which itself is based on the work of Rudnick and Soundararajan. The core technique involves the use of a mollifier to bound the sum from below.

1. Mollifier Construction: The author defines a smoothed Dirichlet polynomial, $L^\dagger_\xi(s)$, approximating the function $\zeta(2s)/\zeta(s)$. This approximation is generated by the Liouville function $\lambda(n)$, which satisfies $\sum \lambda(n)n^{-s} = \zeta(2s)/\zeta(s)$. The polynomial is defined as:
   $$L^\dagger_\xi(s) = \sum_{n \le \xi} \frac{\lambda(n)}{n^s} P\left(\frac{\log(\xi/n)}{\log \xi}\right),$$
   where $\xi = T^\vartheta$ for a fixed $0 < \vartheta < 1$, and $P(x)$ is a polynomial with $P(0)=0$ and $P(1)=1$.

2. Cauchy-Schwarz Inequality: Under the assumption of RH, the symmetry $L^\dagger_\xi(\rho) = L^\dagger_\xi(1-\rho)$ holds for non-trivial zeros. This allows the application of Cauchy's inequality to establish a lower bound:
   $$\sum_{0<\gamma \le T} \left| \frac{\zeta(2\rho)}{\zeta'(\rho)} \right|^2 \ge \frac{|M_1|^2}{M_2},$$
   where $M_1 = \sum_{0<\gamma \le T} \frac{\zeta(2\rho)}{\zeta'(\rho)} L^\dagger_\xi(1-\rho)$ and $M_2 = \sum_{0<\gamma \le T} |L^\dagger_\xi(\rho)|^2$.

3. Contour Integration: To evaluate the asymptotics of $M_1$ and $M_2$, the author employs Cauchy's residue theorem. The sums are converted into contour integrals involving $\zeta(2s)/\zeta(s)$ and $\zeta'(s)/\zeta(s)$ respectively. The contours are rectangular paths shifted to the critical line, and the integrals are split into vertical and horizontal segments.

   • Mean Value Theorems: The evaluation of the main terms relies on the Montgomery-Vaughan mean value theorem for Dirichlet polynomials to handle the oscillatory integrals.
   • Functional Equations: For the left vertical contours, the functional equation of the zeta function is utilized. A key technical distinction from previous work (Milinovich-Ng) is the handling of the $\zeta(2s)$ factor in the numerator, which cancels the decay of $1/\zeta(s)$, requiring an analysis of the oscillation of the ratio $\chi(2s)/\chi(s)$ via the van der Corput second-derivative test.

Key Contributions and Results
The paper establishes the following asymptotic estimates for the moments $M_1$ and $M_2$ (Propositions 5 and 6):

• $M_1 \sim \frac{T}{2\pi} (\log T) \vartheta \int_0^1 P(x) dx$.
• $M_2 \sim \frac{T}{2\pi} (\log T)^2 \left[ \vartheta \int_0^1 P(x)^2 dx + \vartheta^2 \left(\int_0^1 P(x) dx\right)^2 \right]$.

By substituting these estimates into the lower bound inequality and optimizing the choice of the polynomial $P(x)$, the author derives the main theorem:

Theorem 2: Assuming the Riemann Hypothesis and that the non-trivial zeros of $\zeta(s)$ are simple, for any fixed $\epsilon > 0$:
$$\sum_{0<\gamma \le T} \left| \frac{\zeta(2\rho)}{\zeta'(\rho)} \right|^2 \ge \left( \frac{1}{4\pi} - \epsilon \right) T$$
for sufficiently large $T$.

The optimization is achieved by choosing the constant polynomial $P(x) = 1$ and taking the limit as $\vartheta \to 1^-$.

Significance and Claims
The paper claims a rigorous lower bound for the sum in question. The significance of the result lies in its quantitative relationship to Ng's conjecture:

• The proven lower bound is $\frac{1}{4\pi}T$.
• The conjectured asymptotic is $\frac{1}{2\pi}T$.
• Consequently, the result establishes a lower bound that is exactly half the size of the conjectured value.

The author draws a parallel to the work of Milinovich and Ng on Gonek's conjecture (concerning $\sum |\zeta'(\rho)|^{-2}$), where a similar factor-of-two discrepancy exists between the proven lower bound and the conjectured asymptotic. The paper does not claim to prove the full conjecture but provides a non-trivial lower bound that confirms the correct order of magnitude ($O(T)$) while highlighting the difficulty in closing the gap to the conjectured constant.