The Second Moment of Sums of Hecke Eigenvalues I

This paper computes the first and second moments of sums of Hecke eigenvalues for holomorphic cusp forms averaged over large weights, revealing distinct size transitions at $x \approx k$ and $x \approx k^2$ within the regime where the sum length $x$ is smaller than $k^2$.

The Gist

Imagine you are standing in a vast, silent library filled with thousands of unique musical instruments. Each instrument represents a specific mathematical object called a cusp form (a complex wave pattern related to prime numbers and symmetry). When you strike a key on one of these instruments, it doesn't just play a single note; it plays a complex chord made of many frequencies.

In this paper, the author, Ned Carmichael, is trying to understand the volume of these chords when you play them in a specific sequence.

Here is the breakdown of the paper using simple analogies:

1. The Players and the Score

• The Instruments ($f$): Think of each instrument as a unique "song" defined by a number called its weight ($k$). The higher the weight, the more complex and "heavy" the instrument is.
• The Notes ($\lambda_f(n)$): When you play the instrument, it produces a sequence of numbers (eigenvalues). These are like the specific pitches in the chord. Sometimes the pitch is high (positive), sometimes low (negative).
• The Chord Sum ($S(x, f)$): The author isn't interested in just one note. He wants to know what happens if you sum up a block of notes, say from note number $x$ to note number $2x$. Is the total sound loud? Is it quiet? Does it cancel itself out?

2. The Experiment: The "Average" Listener

Since there are thousands of instruments, the author doesn't listen to just one. He listens to all of them at once, but he gives more attention to the "loudest" or most important ones using a special weighting system (called harmonic weights).

He asks two main questions:

1. The First Moment: On average, is the sum of these notes positive, negative, or zero? (Is the chord generally loud or quiet?)
2. The Second Moment: On average, how "energetic" is the chord? (This measures the variance or the "size" of the fluctuations, regardless of whether they are positive or negative).

3. The Big Discovery: The "Phase Transitions"

The most exciting part of the paper is discovering that the behavior of these sums changes dramatically depending on how long the block of notes is (the value $x$) compared to the size of the instrument (the weight $k$).

Think of it like walking through a forest where the trees change type based on how far you've walked:

• Zone 1: The Quiet Zone ($x$ is very small)
  If you only listen to a tiny snippet of the song (where $x$ is much smaller than $k^2$), the notes cancel each other out almost perfectly. The average sound is effectively zero. It's like standing in a soundproof room; the noise is negligible.

• Zone 2: The "Murmuration" Zone (The Transition)
  As you increase the length of the snippet to a specific sweet spot (around $x \approx k^2$), something magical happens. The notes stop canceling out and start syncing up.

  • The Analogy: Imagine a flock of starlings (a murmuration). Individually, they fly randomly. But at a certain density, they suddenly move as one giant, coherent shape.
  • In this zone, the sum of the notes suddenly spikes. The author finds a precise formula for this spike. It's a "phase transition" where the math shifts from silence to a distinct, predictable hum.

• Zone 3: The Oscillatory Zone ($x$ is large)
  If you keep listening past that sweet spot, the behavior changes again. The notes start to oscillate wildly, and the average size of the sum grows, but at a different rate than before.

4. The Secret Weapon: Bessel Functions

How did the author figure this out? He used a mathematical tool called the Petersson Trace Formula.

• The Analogy: Imagine trying to predict the weather by looking at a single cloud. It's impossible. But if you have a formula that tells you how every cloud in the sky interacts with every other cloud, you can predict the storm.
• In this paper, the formula reveals that the interaction between the notes is governed by Bessel Functions.
• The "Peak": Bessel functions are like waves that are usually very small and flat. However, they have a specific "hump" or peak at a certain point.
  • When the length of the note block ($x$) aligns perfectly with the size of the instrument ($k$) so that the Bessel function hits its peak, the sum of the notes explodes in size.
  • When the alignment is off, the Bessel function is flat, and the sum is tiny.

5. Why Does This Matter?

This isn't just about abstract math.

• Predicting the Unpredictable: Number theory often deals with things that look random (like prime numbers). Finding these "transitions" shows that there is hidden order and structure even in the chaos.
• The "Murmuration" Phenomenon: The paper mentions that other mathematicians have seen similar "flocking" behaviors in different contexts. This paper proves that this phenomenon happens specifically when looking at the "weight" of these modular forms.
• Future Work: The author notes that once you go beyond the transition point ($x > k^2$), the sums become surprisingly small again. This suggests a complex, wave-like pattern of growth and decay that mathematicians are just beginning to map out.

Summary

Ned Carmichael's paper is like a study of a massive orchestra. He discovered that if you listen to a specific length of music relative to the size of the orchestra, the sound suddenly swells into a massive, coherent roar (the transition). Before and after that moment, the sound is either a whisper or a chaotic jumble. He used the mathematics of waves (Bessel functions) to pinpoint exactly when that roar happens, revealing a hidden rhythm in the universe of numbers.

Technical Summary

Here is a detailed technical summary of the paper "The Second Moment of Sums of Hecke Eigenvalues I" by Ned Carmichael.

1. Problem Statement

The paper investigates the statistical distribution of sums of Hecke eigenvalues attached to holomorphic cusp forms for the full modular group $SL_2(\mathbb{Z})$. Specifically, let $f$ be a Hecke eigenform of even weight $k$ with Fourier coefficients $\lambda_f(n)$. The author studies the sums:
$$S(x, f) = \sum_{x < n \le 2x} \lambda_f(n)$$
as $f$ varies over an orthogonal basis $B_k$ of cusp forms of weight $k$.

The primary goal is to compute the first and second moments of these sums, averaged over the basis $B_k$ using harmonic weights $\omega(f)$, in the regime where the summation length $x$ is smaller than $k^2$. The paper focuses on identifying and analyzing transitions in the size of these moments as $x$ varies relative to $k$.

2. Methodology

The proofs rely on a combination of analytic number theory tools, specifically:

• Petersson Trace Formula: This is the central tool used to compute the average $\langle \lambda_f(n)\lambda_f(m) \rangle$. It splits the problem into a "diagonal" term (where $n=m$) and an "off-diagonal" term involving Kloosterman sums and Bessel functions $J_{k-1}$.
• Voronoï Summation Formula: Used to transform the sums $S(x, f)$ into dual sums. This is crucial for handling the off-diagonal terms when the Bessel functions are not negligible.
• Poisson Summation: Applied to the dual sums in the second moment calculation to extract the main terms from the off-diagonal contributions.
• Asymptotic Analysis of Bessel Functions: A significant portion of the technical work involves analyzing the behavior of the Bessel function $J_{k-1}(z)$ in three distinct regimes:
  1. Exponential decay: When $z \ll k$.
  2. Transition regime: When $z \approx k$ (specifically $k - k^{1/3} \lesssim z \lesssim k + k^{1/3}$), where the function reaches its maximum. The paper utilizes Airy function approximations here.
  3. Oscillatory regime: When $z \gg k$, where the function oscillates with decaying amplitude.
• Stationary Phase and Integration by Parts: Used to bound oscillatory integrals arising from the Poisson summation and Voronoï transformations.

3. Key Contributions and Results

The paper establishes precise asymptotic formulas for the moments, revealing two distinct transition points where the behavior of the sums changes dramatically.

A. The First Moment $\langle S(x, f) \rangle$

The average size of the sum depends heavily on the relationship between $x$ and $k^2$:

• Regime 1 ($x \lesssim k^2$): If $x \le \frac{k^2}{32\pi^2 + 1}$, the average is exponentially small:
  $$\langle S(x, f) \rangle \ll e^{-\sqrt{k}}$$
  In this range, the off-diagonal terms in the trace formula are negligible because the argument of the Bessel function is too small to reach its peak.
• Regime 2 (Transition): If $x \approx \frac{k^2}{16\pi^2}$ (specifically $\frac{k^2}{32\pi^2 - 1} \le x \le \frac{k^2}{16\pi^2 + 1}$), the Bessel function enters its transition regime. The average becomes significant:
  $$\langle S(x, f) \rangle = (-1)^{k/2} \frac{k}{4\pi} + O(k^{7/8})$$
  This non-zero average is a direct consequence of the "peak" of the Bessel function $J_{k-1}$ aligning with the summation range.

B. The Second Moment $\langle S(x, f)^2 \rangle$

The second moment exhibits a more complex transition structure involving a secondary main term defined by a continuous function $L(\xi)$.

• Regime 1 ($x \lesssim k$): If $x \le \frac{k}{32\pi}$, the off-diagonal terms are negligible, and the sum behaves like a random walk (diagonal dominance):
  $$\langle S(x, f)^2 \rangle = x + O(1)$$
• Regime 2 (Transition): If $x \approx \frac{k}{4\pi}$ (specifically $\frac{k}{32\pi} \le x \le k^{1+\epsilon}$), a secondary main term emerges due to the off-diagonal contribution:
  $$\langle S(x, f)^2 \rangle = x + (-1)^{k/2} \frac{k}{2\pi} L\left(\frac{k}{4\pi x}\right) + O(k^{2/3+\epsilon})$$
  Here, $L(\xi)$ is a piecewise function involving logarithms, non-zero only when $\xi \in [1, 2]$. This corresponds to the range where the Bessel function's transition regime overlaps with the integration domain.
• Regime 3 ($x \gtrsim k$): For larger $x$ (up to $k^2$), the secondary term vanishes, and the error term grows, though the main diagonal term $x$ still dominates.

4. Significance and Context

• Murmurations: The paper connects these transitions to the phenomenon of "murmurations" observed by Bober, Booker, Lee, and Lowry-Duda, where mean values of eigenvalue sums exhibit sudden shifts in behavior at specific scales (here, $x \approx k^2$ for the first moment and $x \approx k$ for the second).
• Analogy with Arithmetic Progressions: The author draws parallels with the distribution of eigenvalues in arithmetic progressions (studied by Lau and Zhao), noting a similar transition at $x \approx q^2$ (where $q$ is the modulus). However, the weight aspect ($k$) presents a unique second transition at $x \approx k$ not present in the level aspect.
• Methodological Advancement: The paper provides a rigorous framework for handling the off-diagonal terms in the Petersson trace formula when the Bessel functions are near their peak. This involves sophisticated applications of Poisson summation and precise asymptotic expansions of Bessel functions using Airy functions.
• Future Work: The results set the stage for subsequent work (cited as [5]) which will analyze the regime where $x > k^2$, where the paper predicts the average size of the sums will drop dramatically (scaling as $x^{1/4}$ rather than $x^{1/2}$).

In summary, this paper provides a comprehensive analysis of the "phase transitions" in the moments of Hecke eigenvalue sums, demonstrating how the interplay between the summation length $x$ and the weight $k$ dictates the statistical behavior of these arithmetic objects through the lens of Bessel function asymptotics.