The Second Moment of Sums of Hecke Eigenvalues II

This paper computes the first and second moments of sums of normalised Hecke eigenvalues over holomorphic cusp forms of large weight, demonstrating that in the range $k^2/(8\pi^2)\leq x\leq k^{12/5-\epsilon}$, the second moment scales between $x^{1/2-o(1)}$ and $x^{1/2}$, a sharp contrast to the linear growth observed in the lower range $x\leq k^{2-o(1)}$.

The Gist

Imagine you are standing in a vast, silent library filled with thousands of unique, magical books. Each book represents a specific mathematical object called a Hecke cusp form. These aren't normal books; they are made of pure numbers and waves.

Inside each book, there is a long list of numbers called Hecke eigenvalues. Think of these numbers as a secret code or a rhythm that the book "sings." Sometimes the numbers are positive, sometimes negative, and they jump around unpredictably.

The author of this paper, Ned Carmichael, is asking a very specific question: If we add up a chunk of these numbers from the middle of the list, how big does the total get?

The Two Worlds of the Library

To understand the paper, imagine the list of numbers in the book is a long road. We are looking at a specific stretch of this road, from mile marker $x$ to mile marker $2x$.

The paper discovers that the behavior of these sums changes dramatically depending on how heavy the book is (mathematically, this is the "weight" $k$) and where on the road we are looking (the value $x$).

1. The "Heavy" Zone (The Old Discovery)

In previous work (referenced as [3] in the paper), mathematicians found that if you look at the road early on (when $x$ is small compared to the book's weight), the sums behave like a random walk.

• The Analogy: Imagine flipping a coin. If you flip it 100 times, the total score might be around 10 or 20. If you flip it 1,000 times, the total score might be around 30 or 40. The size of the sum grows with the square root of the number of flips.
• The Result: In this early zone, the sums are "large" (proportional to $\sqrt{x}$). The numbers don't cancel each other out much; they just pile up.

2. The "Light" Zone (The New Discovery)

This paper investigates what happens when we move further down the road (when $x$ is large, specifically $x \ge k^2/8\pi^2$).

• The Analogy: Imagine you are walking through a dense forest where the wind is blowing in a very specific, complex pattern. If you try to walk in a straight line, the wind pushes you left, then right, then left again.
• The Result: In this new zone, the numbers start to cancel each other out incredibly well. It's like the wind is perfectly organized to push you back to zero. The sum becomes much, much smaller than before. Instead of growing like $\sqrt{x}$, it stays relatively flat, growing only very slowly (like $x^{1/4}$ or even smaller).

The Secret Weapon: The "Bessel Wave"

How did the author figure this out? He used a powerful mathematical tool called the Petersson Trace Formula, which acts like a special pair of glasses.

When you put these glasses on, the messy list of numbers transforms into a different picture involving Bessel functions.

• What is a Bessel function? Think of it as a specific type of wave, like a ripple in a pond.
• The Magic Transition:
  • Before the peak: If the ripple is small compared to the size of the pond, it's weak and quiet.
  • At the peak: The ripple gets huge and loud. This is the "transition zone" where the sums are large.
  • After the peak: Once the ripple passes its peak, it starts to oscillate wildly—up and down, up and down—with the waves getting smaller and smaller.

The paper proves that in the range they are studying, we are deep in the "after the peak" zone. The waves are oscillating so violently and rapidly that when you add them all up, the positive parts cancel out the negative parts almost perfectly.

The "Average" vs. The "Individual"

The paper doesn't just look at one book; it looks at the average behavior of thousands of these magical books.

• The First Moment (The Average): On average, the sums are surprisingly small, almost zero. It's like saying, "If you ask 1,000 people to guess a number, and they all guess randomly, the average guess is the middle."
• The Second Moment (The Variance): This measures how much the sums wiggle around that average. The paper calculates exactly how much they wiggle.
  • In the old zone (small $x$), the wiggle is big.
  • In the new zone (large $x$), the wiggle is much smaller, but it still exists. It's not zero; it's just a tiny, rhythmic vibration.

Why Does This Matter?

In the world of mathematics, these "Hecke eigenvalues" are connected to some of the deepest mysteries in the universe, including the distribution of prime numbers and the shape of the universe itself (via the Langlands program).

Finding out that these sums behave differently in different zones is like discovering that gravity works differently on the Moon than on Earth. It tells us that the underlying structure of these numbers is far more complex and beautiful than we thought. It shows that there is a "phase transition" in the mathematics of these forms, where the chaotic noise suddenly becomes a quiet, organized whisper.

Summary in One Sentence

This paper proves that when you look at the sums of these mysterious number sequences far enough down the line, the chaotic noise cancels itself out due to a specific type of mathematical wave, leaving behind a much smaller, more predictable pattern than anyone expected.

Technical Summary

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

1. Problem Statement

The paper investigates the statistical behavior of sums of normalized Hecke eigenvalues, denoted as $S(x, f)$, for holomorphic cusp forms $f$ of large weight $k$ for the modular group $SL_2(\mathbb{Z})$. Specifically, the author studies:
$$S(x, f) = \sum_{x \le n \le 2x} \lambda_f(n)$$
where $\lambda_f(n)$ are the normalized eigenvalues.

The primary focus is on the first and second moments of these sums, averaged over an orthogonal basis $B_k$ of Hecke eigenforms with respect to the harmonic weight $\omega(f)$. The study targets the regime where the summation length $x$ is large relative to the weight $k$, specifically:
$$\frac{k^2}{8\pi^2} \le x \le k^{12/5 - \epsilon}$$

This regime is of particular interest because it follows a "transition" observed in previous work (Carmichael [3]). In the range $x \le k^{2-o(1)}$, the second moment was shown to be of size $\asymp x$. However, the behavior changes significantly when $x$ exceeds $k^2/(32\pi^2)$, where the Bessel functions governing the spectral expansion enter an oscillatory regime, leading to significant cancellation. The paper aims to quantify this cancellation and determine the precise size of the moments in this "large $x$" regime.

2. Methodology

The proof relies on a combination of analytic number theory tools, specifically:

• Voronoï Summation Formula: The author uses a smoothed version of the Voronoï summation formula (Lemma 2.4) to transform the short sums $S(x, f)$ into dual sums involving Bessel functions. This transformation is crucial for handling the oscillatory nature of the eigenvalues.
• Petersson Trace Formula: To compute the moments (averages over the family $B_k$), the Petersson trace formula (Lemma 2.2) is applied. This splits the second moment into diagonal terms (where indices match) and off-diagonal terms (involving Kloosterman sums).
• Asymptotics of Bessel Functions: A central technical component is the detailed analysis of the Bessel function $J_{k-1}(z)$.
  • For $z < k$, the function is exponentially small.
  • For $z \approx k$, it reaches a peak.
  • For $z > k$, it oscillates with decaying amplitude.
    The paper rigorously utilizes the asymptotic expansion of $J_{k-1}$ in the oscillatory regime (Lemma 2.1) to demonstrate that the integrals in the dual sums exhibit significant cancellation.
• Stationary Phase and Poisson Summation: To bound the off-diagonal terms, the author employs Poisson summation to dualize the sums over $n$. The resulting integrals are estimated using the method of stationary phase (Lemma 2.10), carefully analyzing the phase functions to show that the off-diagonal contribution is negligible compared to the diagonal term in the specified range.
• Equidistribution: To establish a lower bound for the main term (proving it is not zero due to accidental cancellations), the author proves that a specific sequence related to the phase of the Bessel function equidistributes modulo 1 (using the Erdős-Turán inequality and van der Corput's method).

3. Key Contributions and Results

A. Uniform Bound (Theorem 1.1)

The paper establishes a uniform bound for individual sums $S(x, f)$ in the large $x$ regime:
$$S(x, f) \ll x^{1/3+\epsilon}$$
This holds uniformly for all $f \in B_k$ when $x \ge k^2/(8\pi^2)$. This improves upon previous pointwise bounds in this specific range.

B. The First Moment (Theorem 1.2)

The first moment is shown to be of size $x^{1/4}$:
$$\langle S(x, f) \rangle = (-1)^{k/2} \frac{4}{\sqrt{2\pi}} \Omega(1, x) x^{1/4} + O(x^{1/2}k^{-1+\epsilon})$$
where $\Omega(n, x)$ is an explicit oscillatory function derived from the Bessel asymptotics.

C. The Second Moment (Theorem 1.3)

This is the main result. The second moment is shown to be of size $x^{1/2}$, contrasting sharply with the $x$ size found in the small $x$ regime.
$$\langle S(x, f)^2 \rangle = 32\pi x^{1/2} \sum_{n \ge 1} \frac{\Omega(n, x)^2}{n^{3/2}} + O(x^{3/4}k^{-3/5+\epsilon}) + O(k^{29/30+\epsilon})$$
Crucially, the main term satisfies:
$$x^{1/2} \exp\left(-\frac{\log x}{\log \log x}\right) \ll \text{Main Term} \ll x^{1/2}$$
This confirms that the average size of the sum is roughly $x^{1/4}$ (since the second moment is $x^{1/2}$), indicating that the sums are significantly smaller than in the regime $x \le k^2$.

D. The Transition Phenomenon

The paper provides a rigorous explanation for the transition in the size of the sums.

• Small $x$ ($x \lesssim k^2$): The argument of the Bessel function $4\pi\sqrt{nxt}$can be close to the index$k$, hitting the peak of the Bessel function, leading to constructive interference and larger sums ($\sim x^{1/2}$).
• Large $x$ ($x \gtrsim k^2$): The argument $4\pi\sqrt{nxt}$is strictly greater than$k$for all relevant$n$. The Bessel functions are in their oscillatory regime, leading to destructive interference (cancellation) in the integrals, reducing the sum size to$\sim x^{1/4}$.

4. Significance

• Resolution of a Transition: The paper resolves the behavior of Hecke eigenvalue sums in the "large $x$" regime, bridging the gap between the classical mean-square estimates (Chandrasekharan-Narasimhan) and the weight-aspect results. It confirms that the variance drops from order $x$ to order $x^{1/2}$ as $x$ crosses the threshold $k^2/(8\pi^2)$.
• Methodological Advancement: The work demonstrates a sophisticated application of the Voronoï summation formula combined with precise Bessel function asymptotics and stationary phase analysis to handle off-diagonal terms in the weight aspect.
• Connection to L-functions: The techniques used (averaging over weights, handling off-diagonal terms via Poisson summation) are analogous to methods used in the study of moments of $L$-functions (e.g., Hough [8]), suggesting potential applications for central values of $L$-functions in the weight aspect.
• Sharp Contrast: The result highlights a sharp dichotomy in the distribution of Hecke eigenvalues depending on the length of the sum relative to the weight, a phenomenon that was previously only partially understood.

In summary, Carmichael proves that for large weights $k$ and summation lengths $x$ in the range $k^2 \ll x \ll k^{12/5}$, the sums of Hecke eigenvalues are significantly smaller than previously expected for short sums, with the second moment scaling as $x^{1/2}$ rather than $x$. This is driven by the oscillatory nature of the Bessel functions in the spectral expansion.