Generalization on the higher moments of the Fourier coefficients of symmetric power $L$-functions

This paper generalizes and improves upon existing results regarding the asymptotic behavior of the sum of the $l$-th powers of the $n$-th normalized Fourier coefficients of the $j$-th symmetric power $L$-functions for primitive holomorphic cusp forms, specifically for positive integers $l$ and $j$ satisfying $lj \geq 4$.

The Gist

Imagine you are a detective trying to solve a mystery involving a massive, invisible machine. This machine is the Modular Group, and it spits out a long, endless stream of numbers called Fourier coefficients. These numbers aren't random; they follow a very strict, hidden rhythm.

For decades, mathematicians have been trying to understand the "average behavior" of these numbers. Specifically, they wanted to know: If I add up the first $x$ numbers, what do I get? And what happens if I raise these numbers to a high power before adding them?

This paper, written by K. Venkatasubbareddy, is like a detective upgrading their toolkit to solve a specific, tricky version of this mystery. Here is the breakdown using simple analogies:

1. The Characters: The "Symmetric Power" Machine

Think of the original numbers as a melody played by a single instrument (the cusp form $f$).

• Symmetric Powers ($j$): Imagine taking that melody and playing it in harmony with itself. If you play it twice, you get the "2nd symmetric power." If you play it 10 times, you get the "10th symmetric power." Each of these creates a new, more complex melody (a new L-function).
• The Power ($l$): Now, imagine you take a single note from that harmony and shout it out $l$ times (raising it to the power $l$).
• The Goal: The paper looks at the sum of these shouted notes. The authors are interested in cases where the complexity is high (specifically, where the "harmony count" $j$ times the "shout count" $l$ is at least 4).

2. The Problem: The "Noise" in the Data

When you add up these numbers, there is a predictable pattern (the "Main Term"), but there is also a lot of noise (the "Error Term").

• The Main Term: This is the smooth, predictable part of the sum. It's like the steady beat of a drum.
• The Error Term: This is the static, the fuzz, the deviation from the perfect beat. In math, we want this noise to be as small as possible. The smaller the error, the more precise our prediction is.

Previous researchers (like Liu, Luo, and others) had already built a fence around this noise, saying, "The noise is definitely smaller than $x^{0.9}$" (where $x$ is the number of items you are counting). But they couldn't get the fence much lower than that.

3. The Solution: A Sharper Telescope

Venkatasubbareddy's paper is about tightening that fence. The author proves that the noise is actually much smaller than previously thought.

How did they do it?
Think of the numbers as a signal traveling through a foggy tunnel. To see the signal clearly, you need to use a special mathematical tool called Perron's Formula. This tool is like a flashlight that lets you look at the signal from different angles.

• The Old Way: Previous researchers shone their flashlight from a distance. They could see the signal, but the fog (the error) was still thick.
• The New Way: The author found a way to move the flashlight much closer to the signal (moving the "line of integration" in the complex plane).
  • The Catch: Moving the flashlight closer is dangerous. If you get too close, you might hit a "wall" (a mathematical singularity or pole) that ruins the calculation.
  • The Trick: The author realized that for complex cases (where $lj \ge 4$), the "walls" are arranged in a specific way. By using a clever combination of inequalities (like the Cauchy-Schwarz inequality, which is like balancing weights on a scale) and knowing exactly how the "fog" behaves at different distances, they could sneak the flashlight closer without hitting the wall.

4. The Result: A Clearer Picture

Because they could get the flashlight closer, they could see the "Main Term" much more clearly and prove that the "Noise" is significantly smaller.

• The Analogy: Imagine you are trying to hear a whisper in a crowded room.
  • Previous results: "The whisper is audible, but there's a lot of background chatter."
  • This paper: "We found a way to dampen the background chatter so much that the whisper is almost crystal clear."

5. Why Does This Matter?

In the world of number theory, these "Fourier coefficients" are the DNA of prime numbers and deep symmetries in mathematics.

• Precision: Getting a better error term means our mathematical models of the universe are more accurate.
• Generalization: The author didn't just fix one specific case; they created a universal rule that works for any combination of harmony ($j$) and shouting ($l$), as long as the complexity is high enough.

Summary

This paper is a mathematical optimization. The author took a known method for analyzing complex number patterns, realized it was being used too conservatively, and found a way to push the boundaries further. By doing so, they reduced the "error margin" in our understanding of these numbers, giving us a sharper, more precise view of the hidden rhythms of the universe.

In short: They found a better way to filter out the static, revealing a clearer signal in the noise of mathematics.

Technical Summary

Here is a detailed technical summary of the paper "Generalization on the higher moments of the Fourier coefficients of symmetric power L-functions" by K. Venkatasubbareddy.

1. Problem Statement

The paper addresses the problem of estimating the asymptotic behavior of the sum of the $l$-th powers of the normalized Fourier coefficients of the $j$-th symmetric power $L$-functions attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2, \mathbb{Z})$.

Specifically, the author investigates the sum:
$$S_{l,j}(x) = \sum_{n \le x} \lambda_{sym^j f}^l(n)$$
where:

• $f$ is a primitive holomorphic cusp form.
• $\lambda_{sym^j f}(n)$ denotes the $n$-th normalized Fourier coefficient of the $j$-th symmetric power $L$-function, $L(s, sym^j f)$.
• $l$ and $j$ are positive integers.
• The focus is on the case where $lj \ge 4$.

Previous literature had established results for specific small values of $l$ and $j$ (e.g., $l=j=2$, $l=2, j=3,4$, or specific $l$ for $j=2$), often providing error terms of the form $O(x^{\theta + \epsilon})$. The goal of this paper is to generalize these results to arbitrary positive integers $l, j$ (with $lj \ge 4$) and improve the exponents $\theta$ in the error terms compared to recent works by Liu (2023) and Luo et al. (2021).

2. Methodology

The author employs a combination of analytic number theory techniques, specifically:

• Perron's Formula: The sum is converted into a complex contour integral involving the Dirichlet series $L_{l,j}(s) = \sum \lambda_{sym^j f}^l(n) n^{-s}$.
• Decomposition of $L_{l,j}(s)$:
  • Using combinatorial identities (Lemmas 2.1 and 2.2), the author expresses the power $\lambda_{sym^j f}^l(p)$ at a prime $p$ as a linear combination of coefficients of lower-order symmetric power $L$-functions.
  • This allows the Dirichlet series $L_{l,j}(s)$ to be factored into a product of Riemann zeta functions and symmetric power $L$-functions:
    $$L_{l,j}(s) \approx \zeta(s)^{d_{lj/2}} \prod L(s, sym^m f)^{d_m} \times (\text{harmless series})$$
  • The degree of the resulting $L$-function is determined to be $D = (j+1)^l$.
• Contour Shifting: The line of integration is shifted from $\Re(s) = 1+\epsilon$ to $\Re(s) = 1 - \frac{1}{j^3}$.
  • If $lj$ is even, the shift crosses a pole at $s=1$ of order $d_{lj/2}$, generating the main term (a polynomial in $\log x$).
  • If $lj$ is odd, there is no pole at $s=1$, resulting in an error term only.
• Bounds on $L$-functions: The error term estimation relies on:
  • Mean Value Theorems: Using the second moment bounds for general $L$-functions (Perelli).
  • Subconvexity Bounds: Utilizing specific bounds for $\zeta(s)$ (Heath-Brown) and $L(s, sym^2 f)$ (Luo, Rudnick, Soundararajan).
  • Cauchy-Schwarz Inequality: Applied to split the integral of the product of $L$-functions into manageable second moments.
• Optimization: The parameter $T$ (the height of the contour) is optimized to balance the error contributions from the vertical line integral and the horizontal segments.

3. Key Contributions

A. Generalization

The paper extends previous results which were limited to specific pairs of $(l, j)$ to a general framework for all positive integers $l, j$ such that $lj \ge 4$.

B. Improved Error Exponents

The author derives new exponents $\theta_{l,j}$ for the error term $O(x^{\theta_{l,j} + \epsilon})$. These exponents are strictly smaller (better) than those previously obtained by Liu (2023).

The main result (Theorem 1) states:
$$\sum_{n \le x} \lambda_{sym^j f}^l(n) = \begin{cases} x P_{d_{lj/2}-1}(\log x) + O(x^{\theta_{l,j} + \epsilon}) & \text{if } lj \text{ is even} \\ O(x^{\theta_{l,j} + \epsilon}) & \text{if } lj \text{ is odd} \end{cases}$$

Where $P_k$ is a polynomial of degree $k$, and the exponent $\theta_{l,j}$ is defined as:

• Case $lj = 4$:
  $$\theta_{l,j} = 1 - \frac{126 j^{3/2}}{63(D-d_2)j^{3/2} + 16\sqrt{15}d_2}$$
• Case $lj \ge 6$ (even):
  $$\theta_{l,j} = 1 - \frac{630 j^{3/2}}{j^{3/2}(315D - 315d_{lj/2} - 189d_{lj/2-1}) + 80\sqrt{15}d_{lj/2}}$$
• Case $lj \ge 5$ (odd):
  $$\theta_{l,j} = 1 - \frac{6}{3D - 2e_{(lj-1)/2}}$$

Here, $D = (j+1)^l$ is the degree of the $L$-function, and $d_m, e_m$ are coefficients derived from the combinatorial decomposition of the power of the Fourier coefficients.

C. Refined Estimates

The paper also discusses a potential further refinement of the error term by moving the integration line to $\Re(s) = 1 - \sigma(j)$ with $\sigma(j) < 1/j^3$, yielding slightly better exponents $\theta^*_{l,j}$ (detailed in Remark 1.2 and the provided tables).

4. Key Results and Data

The author provides a comparative table showing the improvement over previous exponents (from Liu [15] and Luo et al. [16]).

Examples of Improvement ($j=2$):

• $l=2$ ($lj=4$): Previous $\approx 0.7642$ $\to$ New $\theta \approx 0.7604$ $\to$ Refined $\approx 0.75$.
• $l=3$ ($lj=6$): Previous $\approx 0.9193$ $\to$ New $\theta \approx 0.9185$.
• $l=8$ ($lj=16$): Previous $\approx 0.9996868$ $\to$ New $\theta \approx 0.9996852$.

Examples of Improvement ($l=2$):

• $j=3$ ($lj=6$): Previous $\approx 0.866$ $\to$ New $\theta \approx 0.8629$.
• $j=8$ ($lj=16$): Previous $\approx 0.97499$ $\to$ New $\theta \approx 0.97482$.

5. Significance and Limitations

Significance:

• Advancement in Analytic Number Theory: The paper pushes the boundaries of understanding the distribution of Fourier coefficients of symmetric power $L$-functions, a central topic in the Langlands program and automorphic forms.
• Technical Refinement: By optimizing the choice of the integration line and utilizing specific subconvexity bounds more effectively, the author achieves tighter error bounds, which are crucial for applications requiring precise asymptotic estimates.
• Unified Framework: The derivation of a general formula for arbitrary $l$ and $j$ (subject to $lj \ge 4$) unifies scattered results in the literature.

Limitations:

• Constraint $lj \ge 4$: As noted in Remark 1.1, the method fails for $lj \le 3$. In these cases, the decomposition of the $L$-function $L_{l,j}(s)$ contains at most two factors. This prevents the application of the specific subconvexity splitting used in the proof, forcing reliance on Cauchy-Schwarz or Hölder inequalities which result in a "loss" (worse error terms) that the current method cannot overcome.
• Dependence on Known Bounds: The quality of the result relies on the current state-of-the-art subconvexity bounds for $L(s, sym^2 f)$ and $\zeta(s)$. Future improvements in subconvexity would directly improve the exponents $\theta_{l,j}$ derived here.

In conclusion, this paper represents a significant step forward in the quantitative analysis of moments of symmetric power $L$-functions, offering a generalized and improved framework for estimating these sums.