Landau–Gonek Theorem Overview

• The Landau–Gonek Theorem is a precise asymptotic result for sums of the form X^ρ over the nontrivial zeros of the Riemann zeta function, highlighting its sensitivity to prime power inputs.
• It uses complex-analytic techniques such as contour integration, residue calculus, and stationary-phase analysis to uniformly handle different regimes of X relative to T.
• Applications include explicit evaluations of discrete zeta function moments and refined zero statistics, underpinning results in prime number theory and Shanks’ conjecture.

The Landau–Gonek Theorem provides a precise asymptotic for sums of the form $\sum_{0<\gamma\le T} X^\rho$ over the nontrivial zeros $\rho = \beta + i \gamma$ of the Riemann zeta function $\zeta(s)$. The theorem quantifies both the deep arithmetic sensitivity of such sums and enables applications to the discrete moments of $\zeta$ and its derivatives, with significant consequences for the understanding of zero statistics and explicit formulas in prime number theory (Durkan et al., 25 Jan 2026, Aryan, 2019).

1. Classical Landau–Gonek Theorem

The classical result examines sums of $X^\rho$ for real $X > 0$ as $T \to \infty$. Landau (1911) originally showed that: $$\sum_{0<\gamma\le T} X^\rho = -\frac{T}{2\pi}\,\Lambda(X) + O(\log T),$$ where $\Lambda(n)$ is the von Mangoldt function. This main term is nonzero only if $X$ is a prime power, revealing strong arithmetic sensitivity.

Gonek (1985) extended the result uniformly for $X, T > 1$: $$\sum_{0<\gamma\le T} X^\rho = -\frac{T}{2\pi} \Lambda(X) + O\!\bigl(X\log(2XT)\log\log(3X)\bigr) + O\!\left(\log X\,\min(T, \tfrac{X}{\langle X\rangle})\right) + O\!\left(\log(2T)\,\min(T, \tfrac{1}{\log X})\right),$$ where $\langle X\rangle$ is the distance from $X$ to the nearest prime power unequal to $X$ (Durkan et al., 25 Jan 2026, Aryan, 2019).

2. Extensions and Oscillatory Generalisations

Recent developments generalise the Landau–Gonek formula by inserting the oscillatory factor $\chi(\rho)$, the root term from the functional equation of $\zeta$: $\zeta(s) = \chi(s)\,\zeta(1-s).$ Specifically, the sum

$S(X,T) = \sum_{T<\gamma\le2T} \chi(\rho)\,X^\rho$

splits into three distinct arithmetic regimes in $X$ relative to $T$, each described by different asymptotic forms and principal error terms.

Regime	Main Term Shape	Transition Points
$X\leq \frac{T}{2\pi}$	$$-\!X\sum_{\frac{T}{2\pi X}<n\le\frac{T}{\pi X}}\Lambda(n)e^{2\pi i X n}$$	$X = T/(2\pi)$
$\frac{T}{2\pi}<X\leq \frac{T}{\pi}$	$X(\log X)e^{2\pi i X}$	$X = T/\pi$
$X>\frac{T}{\pi}$	$$-\!X\sum_{\frac{\pi X}{T}\le n<\frac{2\pi X}{T}}\frac{\Lambda(n)}{n}e^{2\pi i X/n}$$	—

These regimes arise from stationary-phase analysis of oscillatory integrals, and the transitions at $X = T/(2\pi)$ and $X = T/\pi$ are absorbed by the principal error term (Durkan et al., 25 Jan 2026).

3. Contour Integration and Stationary-Phase Analysis

The proofs employ complex-analytic methods, notably Cauchy's residue theorem to convert sums over zeros into contour integrals involving $\zeta'/\zeta$, Dirichlet series expansions, and explicit handling of the functional equation via $\chi(s)$. The stationary-phase method determines principal contributions from regions where oscillatory integrals do not rapidly cancel, delineating cases where main terms or errors dominate.

For example, integrals of the form

$$I_1 = -\frac{1}{2\pi}\sum_{n\ge1} \Lambda(n)\left(\frac{X}{n}\right)^c \int_T^{2T}\chi(c+it)\left(\frac{X}{n}\right)^{it}dt$$

are analyzed by identifying the $2\pi X/n \in (T, 2T)$ region for stationary-phase, yielding the dominant term in $S(X,T)$ (Durkan et al., 25 Jan 2026).

4. Extensions: Local Correlation Weights and Pair Correlation Estimates

Aryan's extension introduces local-correlation weights, notably Gaussian and Fejér-type Dirichlet kernels, leading to the following weighted sum: $$2^{3/2} M \sum_{\rho,\,\rho'} w(\rho')\,e^{-M^2(\rho'-\rho)^2} x^\rho,$$ where $w(s)$ is a Gaussian weight centered at $1/2 + iT$ and $M = a\sqrt{\log T}$ for $0$x = p$ or $x = pq$ ($p,q$ prime), the main term persists (up to a harmless Gaussian factor). For other $x$, the sum is bounded by $O(1)$ (Aryan, 2019).

When the Gaussian is replaced by a Dirichlet kernel $W_\rho(s)$, one obtains unconditional Montgomery-type pair correlation: $$\sum_{T/2<\gamma,\gamma'<T} K_a\left(\frac{(\gamma-\gamma')\log T}{2\pi}\right) = \frac{T\log T}{2\pi} \int_{-a}^a\left(1-\frac{|u|}{a}\right) du + O(T \log^2 T),$$ where $K_a(u)$ is the Fejér kernel (Aryan, 2019).

5. Arithmetic Sensitivity and Discrete Moments of the Zeta Function

The theorems depend critically on whether $X$ is an integer or sufficiently close to a prime power—the phase factors $e^{2 \pi i X n}$ or $e^{2 \pi i X/n}$ result in massive cancellation unless stationary, ensuring main terms only when arithmetic conditions are met. This sensitivity feeds directly into calculating discrete moments via the approximate functional equation: $$\zeta'(s) = -\sum_{m\le t^\alpha} \frac{\log m}{m^s} + \chi(s) \sum_{n\le t^{1-\alpha}} \frac{\log n - \log(t/2\pi)}{n^{1-s}} + O(\dots),$$ allowing for explicit evaluation of moments such as: $$\sum_{0<\gamma\le T} \zeta'(\rho) = \frac{T}{4\pi}\left(\log \frac{T}{2\pi}\right)^2 + (\gamma_0 - 1)\frac{T}{2\pi}\log\left(\frac{T}{2\pi}\right) + (1 - \gamma_0 - \gamma_0^2 - 3\gamma_1)\frac{T}{2\pi} + O\left(T\,e^{-c\sqrt{\log T}}\right).$$ This recovers (unconditionally) the full asymptotic predicted by Shanks' conjecture and its generalisations (Durkan et al., 25 Jan 2026).

6. Consequences for Zero Simplicity and Pair Correlation

Using the extended Landau–Gonek formula with local-correlation weights, unconditional pair correlation estimates are obtained—these match the triangular kernel form of Montgomery's RH-conditional result but without assuming the Riemann Hypothesis. Under a standard zero-density hypothesis

$$N(\sigma, T) \ll T^{2(1-\sigma)} (\log T)^{-B}, \quad B > 4,$$

it follows combinatorially (via test function insertion and Fejér kernel smoothing) that at least two-thirds of the nontrivial zeros of $\zeta(s)$ are simple. Zeros off the critical line contribute negligibly due to exponential decay in their smoothing weights, and only simple zeros account for the main mass (Aryan, 2019).

7. Applications: Shanks’ Conjecture and Higher-Derivative Moments

Shanks’ conjecture asserts positivity of the discrete average $\sum_{0<\gamma\le T} \zeta'(\rho)$ for large $T$. The generalised Landau–Gonek theorem provides a conceptually unified proof independent of RH, substantiating

$\sum_{0<\gamma\le T} \zeta'(\rho) > 0$

for large $T$. Likewise, for higher derivatives, the explicit asymptotic

$$\sum_{0<\gamma\le T} \zeta^{(\nu)}(\rho) = \frac{(-1)^{\nu + 1} (\nu + 1) T}{2\pi} \left(\log \frac{T}{2\pi}\right)^{\nu + 1} + O(T(\log T)^\nu)$$

holds for each $\nu \ge 0$ (Durkan et al., 25 Jan 2026). This framework applies to mean-value analyses for the Riemann zeta function and sharpens the explicit connection between zero statistics, prime number theory, and analytic number theory.