The Lindelöf Hypothesis for Zeta Zero Ordinates

1. Problem Statement and Motivation

The paper investigates the distribution of the nontrivial zeros $\rho = \beta + i\gamma$ of the Riemann zeta-function $\zeta(s)$. Specifically, it addresses a conjecture proposed by Gonek, Graham, and Lee regarding the Lindelöf Hypothesis for the ordinates of these zeros (denoted as $LH(\gamma)$).

The core problem is to estimate the exponential sum:
$$\sum_{0 < \gamma < x} \gamma^{-i\tau}$$
where the summation runs over the imaginary parts (ordinates) of the nontrivial zeros. The conjecture posits that this sum behaves asymptotically like an integral, with a specific error term bound:
$$\sum_{0 < \gamma < x} \gamma^{-i\tau} = \int_{2}^{x} \frac{u^{-i\tau}}{2\pi \log u} du + E$$
where the error term $E$ satisfies $E \ll |\tau|^\epsilon x^{1/2}$ for $2 \le x \le |\tau|^B$.

This is analogous to the classical Lindelöf Hypothesis for $\zeta(1/2+it)$, which concerns the growth of the zeta function itself. The authors aim to prove this asymptotic formula unconditionally for small $|\tau|$ and conditionally (assuming the Riemann Hypothesis) for larger ranges, and to establish an equivalence between the error term of this sum and the growth of a specific Dirichlet series.

2. Methodology

The authors employ a combination of analytic number theory techniques, including:

• Stieltjes Integration: Used to relate the sum over zeros to the Riemann-von Mangoldt counting function $N(T)$ and the argument function $S(T)$.
• Contour Integration: The residue theorem is applied to integrals involving the logarithmic derivative $\zeta'/\zeta(s)$ and the functional equation of the zeta function.
• Stationary Phase Method & Derivative Tests: Used to estimate oscillatory integrals arising from the Dirichlet series representation of $\zeta'/\zeta$.
• Perron's Formula: A truncated version is utilized to express the partial sum of the Dirichlet series $G(s) = \sum \gamma^{-s}$ in terms of a complex integral.
• Analytic Continuation: The properties of the Dirichlet series $G(s)$ are analyzed to link its vertical growth to the error term in the zero ordinate sum.

3. Key Contributions and Results

Theorem 1: Unconditional Result for Small $\tau$

The authors prove an unconditional asymptotic formula for the sum $\sum_{0 < \gamma < x} \gamma^{-i\tau}$.

• Result: For any real $\tau$ and $x > 0$:
  $$\sum_{0 < \gamma < x} \gamma^{-i\tau} = \frac{x^{1-i\tau}}{2\pi(1-i\tau)} \left( \log \frac{x}{2\pi} - \frac{1}{1-i\tau} \right) + O(|\tau|(\log x)^2)$$
• Significance: This confirms the Lindelöf Hypothesis for ordinates ($LH(\gamma)$) in the range where $|\tau|^2 \le x \le \tau^B$. In this range, the main term dominates the error term. The proof relies on the classical bound $S(T) \ll \log T$.

Theorem 2: Contour Integration and Conditional Results

This theorem provides a more refined analysis using contour integration, splitting the problem into two cases based on the magnitude of $\tau$ relative to $x$.

• Case 1 (Small $\tau$): For $0 \le \tau \le cx \log 2$, the sum is approximated by the integral with an error of $O((\log x)^2)$.
• Case 2 (Large $\tau$): For $cx \log 2 \le \tau \ll x \log x$, the sum transforms into an exponential sum involving the von Mangoldt function $\Lambda(n)$:
  $$\sum_{x \le \gamma < y} \gamma^{-i\tau} \approx \frac{e^{i(\tau+\pi/4)}\tau^{1/2-i\tau}}{(2\pi)^{1/2}} \sum_{e^{\tau/y} \le n \le e^{\tau/x}} \frac{\Lambda(n)(\log n)^{i\tau}}{n^{1/2}\log n} + \text{Error}$$
• Corollary 1 (Conditional on RH): Assuming the Riemann Hypothesis (RH), the authors derive bounds for the sum. Specifically, if $x \le \gamma < 2x$:
  • For small $\tau$: The sum is bounded by $O(x \log x / |\tau| + (\log x)^2)$.
  • For large $\tau$: The sum exhibits "square root cancellation," behaving as $x^{1/2+o(1)}$. This suggests that if RH is true, the Lindelöf Hypothesis for ordinates holds in the difficult range where $x \approx |\tau|^2$.

Theorem 3: Equivalence with Dirichlet Series Growth

The paper establishes a profound equivalence between the error term of the zero ordinate sum and the growth of the Dirichlet series $G(s) = \sum_{\gamma > 0} \gamma^{-s}$.

• Statement: Let $A \in [0, 1/2]$. The following are equivalent:
  1. The error term $E$ in the asymptotic formula satisfies $E \ll |\tau|^{A+\epsilon} x^{1/2}$ for $2 \le x \le |\tau|^2$.
  2. The Dirichlet series satisfies the vertical bound $G(1/2 + it) \ll |t|^{A+\epsilon}$ for $|t| \ge 1$.
• Significance: This connects the distribution of zeros (via the sum) directly to the analytic properties of the series $G(s)$. It implies that proving the Lindelöf Hypothesis for ordinates is equivalent to proving a specific growth bound for $G(s)$ on the critical line.

4. Significance and Impact

• New Perspective: The paper introduces a new perspective on the Lindelöf Hypothesis, shifting focus from the values of $\zeta(s)$ to the distribution of its zeros' ordinates.
• Unconditional Progress: It provides the first unconditional proof of the conjecture in the range $|\tau|^2 \le x$, a regime where the main term dominates.
• Conditional Optimality: Under the assumption of the Riemann Hypothesis, the results suggest that the "square root cancellation" ($x^{1/2}$) is the optimal behavior for these sums, mirroring the classical Lindelöf Hypothesis.
• Analytic Connection: The equivalence in Theorem 3 offers a new pathway for research. Instead of attacking the sum directly, one could study the analytic continuation and growth of the Dirichlet series $G(s)$, which has been studied by Delsarte, Ivić, and Bondarenko.
• Historical Context: The work bridges classical results (von Mangoldt, Littlewood) with modern conjectures (Gonek et al.), clarifying the relationship between the argument of the zeta function ($S(T)$) and the distribution of zeros.

In summary, the paper rigorously quantifies the behavior of exponential sums over zeta zeros, proving the Lindelöf Hypothesis for ordinates in specific ranges and establishing a deep theoretical link between these sums and the growth of associated Dirichlet series.