---
title: Landau–Gonek Theorem Overview
url: https://www.emergentmind.com/topics/landau-gonek-theorem
type: topic
---

# Landau–Gonek Theorem Overview

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 [2601.18025], [1902.05473].

## 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 $X$ [2601.18025], [1902.05473].

## 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 [2601.18025].

## 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 $S(X,T)$ [2601.18025].

## 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 $0<a<1/2$. For $x = p$ or $x = pq$ ($p,q$ prime), the main term persists (up to a harmless Gaussian factor). For other $x$, the sum is bounded by $O(1)$ [1902.05473].

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 [1902.05473].

## 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 [2601.18025].

## 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 [1902.05473].

## 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 $\nu \ge 0$ [2601.18025]. 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.

Source: https://www.emergentmind.com/topics/landau-gonek-theorem