Note on the positivity of the real part of the log-derivative of the Riemann $ξ$-function near the critical line

Technical Summary: Positivity of the Real Part of the Log-Derivative of the Riemann $\xi$-Function

Problem Statement
The paper investigates the behavior of the real part of the logarithmic derivative of the Riemann $\xi$-function, denoted as $\Re \frac{\xi'}{\xi}(s)$, within the critical strip $0 < \sigma < 1$. Specifically, the authors focus on the region immediately to the right of the critical line, defined by $1/2 + 1/\sqrt{\log t} < \sigma < 1$ for sufficiently large $t$.

While it is known that $\Re \frac{\xi'}{\xi}(s) > 0$ for $\sigma > 1$ and that the Riemann Hypothesis (RH) is equivalent to this positivity holding for all $\sigma > 1/2$, the behavior in the strip $1/2 < \sigma < 1$ remains a subject of study. Previous work by Goldstein and A.G. [5] established a lower bound for the sum over zeros on the critical line, but this bound tended to zero as $\sigma \to 1/2$. The primary objective of this work is to improve this lower bound to one that grows infinitely as $\sigma \to 1/2$, thereby providing a stronger condition for positivity in the vicinity of the critical line.

Methodology
The authors employ analytic number theory techniques, primarily relying on the infinite product representation of the $\xi$-function and partial summation methods.

1. Logarithmic Derivative Expansion: The study utilizes the expansion $\frac{\xi'}{\xi}(s) = \sum_{\rho} \frac{1}{s-\rho}$, where the sum runs over the non-trivial zeros $\rho$ of the Riemann zeta function. The real part of this sum is decomposed into contributions from zeros on the critical line ($\rho = 1/2 + i\gamma$) and hypothetical zeros off the critical line ($\tilde{\rho} = \tilde{\beta} + i\tilde{\gamma}$).
2. Partial Summation and Integral Approximation: To estimate the sum over critical line zeros, the authors utilize partial summation (Lemma 2) combined with explicit bounds for the zero-counting function $N(T)$ (Lemma 3).
3. Integral Estimation: A significant portion of the technical work involves deriving explicit lower bounds for integrals of the form $\int \frac{\log u}{a^2(t) + b^2(t)(u-t)^2} du$. Lemma 5 provides a rigorous lower bound for these integrals, which are used to approximate the discrete sums over zeros.
4. Handling Hypothetical Zeros: The paper analyzes scenarios where RH fails by considering the contribution of zeros off the critical line. Lemma 9 provides an upper bound for the sum of inverse squares of distances to hypothetical zeros, ensuring their contribution does not overwhelm the positive contribution from the critical line zeros in the specified region.

Key Contributions and Results

• Improved Lower Bound (Theorem 6): The central result is Theorem 6, which establishes a new explicit lower bound for the real part of the sum over critical line zeros. For $1/2 + 1/\sqrt{\log t} < \sigma < 1$ and $t > 3.11 \times 10^{10}$, the authors prove:
  $$0 < (0.28 - \epsilon(t))c \frac{1}{\sigma - 1/2} < \Re \sum_{\rho=1/2+i\gamma} \frac{1}{s-\rho}$$
  Here, $c$ represents the lower density of zeros on the critical line ($c = \liminf N_{1/2}(T)/N(T)$), and $\epsilon(t)$ is a function vanishing as $\frac{\log \log t}{\log t}$. Unlike previous bounds, this expression diverges as $\sigma \to 1/2$, offering a stronger constraint.

• Analysis of RH Failure Scenarios: Section 4 investigates the positivity of $\Re \frac{\xi'}{\xi}(s)$ under the assumption that RH is false. The authors consider three cases:

  1. A single zero off the critical line.
  2. A finite number of zeros off the critical line.
  3. Infinitely many zeros off the critical line.
     In all cases, the authors demonstrate that $\Re \frac{\xi'}{\xi}(s)$ remains positive in the region $1/2 + 1/\sqrt{\log t} < \sigma < 1$, except potentially in small neighborhoods surrounding the hypothetical non-trivial zeros. The positive contribution from the critical line zeros (scaled by $c$) dominates the negative contributions from off-line zeros in the specified region.

• Numerical Verification: The authors use the Odlyzko database [12] to visualize the region where the inequality holds, confirming the theoretical bounds for specific large values of $t$ (e.g., $t \approx 2.68 \times 10^{11}$).

Significance and Claims
The paper claims to refine the understanding of the $\xi$-function's log-derivative near the critical line. By obtaining a lower bound that grows as $\sigma \to 1/2$, the work strengthens the evidence for the positivity of $\Re \frac{\xi'}{\xi}(s)$ in the critical strip, even without assuming the Riemann Hypothesis is true.

The authors modestly frame their contribution as an improvement upon the results of Goldstein and A.G. [5]. They do not claim to prove the Riemann Hypothesis; rather, they show that if RH fails, the "positivity region" of the log-derivative is robust, persisting near the critical line except in the immediate vicinity of hypothetical counter-examples. The work provides explicit, computable bounds that depend on the density of critical line zeros ($c$) and the magnitude of $t$.