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Moments and sign changes of symmetric power $L$ -function coefficients over sums of squares

This paper establishes upper bounds for partial sums and asymptotic formulas for squared coefficients of symmetric power $L$ -functions over integers represented as sums of $m$ squares (for $2 \le m \le 12$ ), which are then applied to derive lower bounds for the number of sign changes of these coefficients along such sequences.

Original authors: Jewel Mahajan, Arnab Mitra

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jewel Mahajan, Arnab Mitra

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing in a vast, infinite library. Inside this library, there is a special book called a Hecke eigenform. This book isn't made of words, but of numbers. Every page of the book contains a specific number, and these numbers follow a very strict, rhythmic pattern. Mathematicians call these numbers coefficients.

Now, imagine you want to study these numbers, but looking at them one by one is too slow and chaotic. So, you decide to group them. You create a special filter called a Symmetric Power L-function. Think of this filter as a lens that takes the original numbers and rearranges them into a new, more complex sequence. Let's call the numbers in this new sequence $\lambda$ .

The authors of this paper, Jewel Mahajan and Arnab Mitra, are interested in two main things about these $\lambda$ numbers:

How big are they when you add them up? (The "Moments")
Do they flip between positive and negative? (The "Sign Changes")

Here is the twist: They aren't just adding up any numbers. They are only adding up the numbers that correspond to sums of squares.

The "Sum of Squares" Game

Imagine you have a pile of blocks. You want to build a square tower using these blocks.

2 Squares: You can build a tower if the total number of blocks can be written as $a^2 + b^2$ (like $1^2 + 2^2 = 5$ ).
4 Squares: You can build a tower if the blocks equal $a^2 + b^2 + c^2 + d^2$ .
The paper looks at towers built with 2, 4, 6, 8, 10, and 12 square layers.

The authors ask: "If we only look at the $\lambda$ numbers that fit into these specific square-tower shapes, what happens?"

The Findings: The "Weather Report" of Numbers

1. The Size of the Storm (Upper Bounds)

The first part of the paper is like a weather forecast. The authors want to know: "If we sum up all these $\lambda$ numbers up to a certain size $x$ , how big can the total get?"

The Result: They found that the total sum doesn't explode to infinity. It grows, but it stays within a very specific, predictable limit.
The Analogy: Imagine a river flowing. You might think the water level could rise forever, but the authors proved that the river is actually contained within a very narrow, steep canyon. No matter how far you go downstream (how large $x$ gets), the water level (the sum) never exceeds a certain height. They calculated exactly how high that water level can get for different numbers of square layers (2, 4, 6, etc.).

2. The Average Rainfall (Asymptotic Formulas)

Next, they looked at the squares of these numbers ( $\lambda^2$ ). Since squaring a number always makes it positive, this is like measuring the total "energy" or "intensity" of the numbers.

The Result: When they added up the squared numbers for these specific square-tower shapes, they found a clear, steady pattern. The total energy grows in a straight line (or a smooth curve) that is easy to predict.
The Analogy: If the first result was about the height of the water, this result is about the total volume of water that has passed through. They found that for 2, 4, 6, 8, 10, and 12 square layers, the volume of water follows a precise formula. It's like knowing that for every mile you drive, you use exactly 3 gallons of gas. They gave the exact "gallons per mile" for these mathematical sequences.

3. The Flip-Flop Dance (Sign Changes)

Finally, the most exciting part: Do these numbers change their "mood"? Do they switch from being happy (positive) to sad (negative)?

The Problem: If a sequence of numbers stays positive for a long time, it's boring. If it flips back and forth wildly, it's chaotic. Mathematicians want to know: How often does it flip?
The Result: The authors proved that these numbers do flip. In fact, they flip a lot.
The Analogy: Imagine a pendulum swinging. The authors proved that between any two large distances (say, between mile 1,000 and mile 2,000), this pendulum swings back and forth a huge number of times.
- For 2 squares, they proved the numbers flip at least $x^{1-\delta}$ times.
- For 4, 6, 8, 10, and 12 squares, they proved the numbers flip even more frequently (or at least as frequently, depending on the specific math).
- Essentially, they showed that these numbers are never "stuck" in one mood for very long. They are constantly dancing between positive and negative.

Why Does This Matter?

In the world of pure mathematics, these numbers are like the DNA of the universe's symmetry. By understanding how they behave when filtered through "sums of squares," the authors are mapping out the hidden structure of these mathematical objects.

They didn't just guess; they built a rigorous mathematical bridge using tools like Perron's formula (a way to count things using complex numbers) and residue theory (finding the "peaks" in a mathematical landscape).

In summary:
The paper takes a complex, invisible sequence of numbers, filters them through the lens of "sums of squares" (2, 4, 6... 12), and proves three things:

The total sum stays within a predictable, small limit.
The total "energy" (squared sum) follows a smooth, predictable growth curve.
The numbers constantly flip between positive and negative, proving they are full of dynamic activity rather than being static.

This is a map of the rhythm and structure of a very abstract part of mathematics, showing us that even in the most complex number patterns, there is order and constant motion.

Technical Summary: Moments and Sign Changes of Symmetric Power L-Function Coefficients Over Sums of Squares

Problem Statement
This paper investigates the analytic properties of the Dirichlet coefficients, denoted $\lambda_{\text{sym}^j f}(n)$ , of the $j$ -th symmetric power $L$ -function attached to a normalized Hecke eigenform $f$ of even integral weight for the full modular group $SL(2, \mathbb{Z})$ . Specifically, the authors focus on the behavior of these coefficients when the argument $n$ is restricted to integers representable as a sum of $m$ squares, where $m$ is an even integer in the range $2 \le m \le 12$ . The study addresses two primary objectives: establishing upper bounds for the partial sums of $\lambda_{\text{sym}^j f}(n)$ and deriving asymptotic formulas for the second moments $\lambda_{\text{sym}^j f}^2(n)$ over these sequences. Furthermore, the paper utilizes these moment estimates to derive lower bounds for the number of sign changes of the coefficients within intervals $[x, 2x]$ .

Methodology
The proofs rely on classical analytic number theory techniques, primarily Perron's formula, to express partial sums as contour integrals of associated Dirichlet series. The core of the methodology involves the factorization of these Dirichlet series into products of $L$ -functions and absolutely convergent Dirichlet series.

Arithmetic Decomposition: The authors utilize explicit formulas for the number of representations of an integer as a sum of $m$ squares, $r_m(n)$ . These formulas express $r_m(n)$ as linear combinations of divisor functions and Dirichlet characters (specifically the non-principal character $\chi_4$ modulo 4). This allows the sums over $m$ squares to be rewritten as sums of the form $\sum_{n \le x} \lambda_{\text{sym}^j f}(n) \cdot \text{arithmetic\_function}(n)$ .
Dirichlet Series Factorization: A series of lemmas (Lemmas 14–30) establish that the Dirichlet series associated with $\lambda_{\text{sym}^j f}(n)$ twisted by these arithmetic functions admit factorizations of the form $G(s)H(s)$ . Here, $G(s)$ consists of products of symmetric power $L$ -functions (and their twists by $\chi_4$ ) and Riemann zeta functions, while $H(s)$ is a Dirichlet series converging absolutely in a half-plane extending to the left of the critical line.
Contour Integration and Bounds: The authors shift the line of integration in Perron's formula to the left, picking up residues at poles (which yield the main terms for the second moments) and bounding the remaining integrals. The error terms are controlled using convexity bounds and mean value estimates for $L$ -functions (Lemmas 31–34), including bounds for the Riemann zeta function and symmetric power $L$ -functions in the critical strip.
Sign Change Analysis: To estimate the number of sign changes, the authors employ a contradiction argument. They assume the sequence maintains a constant sign over a short interval $[x, x+h]$ . By comparing the upper bound of the first moment (derived from the partial sum estimates) with the lower bound of the second moment (derived from the asymptotic formulas), they demonstrate that the assumption leads to a contradiction for sufficiently large $x$ , thereby proving the existence of sign changes.

Key Contributions and Results

Partial Sums (First Moments): The paper establishes upper bounds for the partial sums of $\lambda_{\text{sym}^j f}(n)$ over sums of $m$ squares.
- For $m=2$ , the sum is bounded by $O(x^{1 - \frac{1}{j+1} + \epsilon})$ .
- For $m \in \{4, 6, 8, 10, 12\}$ , the sum is bounded by $O(x^{\frac{m}{2} - \frac{2}{j+3} + \epsilon})$ .
- These results generalize previous work on specific cases (e.g., sums of 4 or 6 squares) to a broader range of $m$ and $j$ .
Second Moments: The authors derive asymptotic formulas for the sums of squares of the coefficients, $\sum \lambda_{\text{sym}^j f}^2(n)$ , over sums of $m$ squares.
- For $m=2$ , the sum is $c_{1,j,f}x + O(x^{1 - \frac{1}{(j+1)^2} + \epsilon})$ .
- For $m \in \{4, 6, 8, 10, 12\}$ , the sum is $c_{m,j,f}x^{\frac{m}{2}} + O(x^{\frac{m}{2} - \frac{2}{(j+1)^2} + \epsilon})$ .
- The main terms involve effective constants depending on $j, f,$ and $m$ , expressed via products of $L$ -values at specific points.
Sign Changes: Applying the moment estimates, the paper proves lower bounds for the number of sign changes of the sequence $\{\lambda_{\text{sym}^j f}(n)\}$ restricted to sums of squares.
- For sums of two squares, the sequence has at least $x^{1-\delta_j}$ sign changes in $[x, 2x]$ for any $\delta_j$ satisfying $1 - \frac{1}{(j+1)^2} < \delta_j < 1$ . This represents a slight improvement over a previous result by Kaur (2026) for the sum of two squares.
- For sums of $m$ squares where $m \in \{4, 6, 8, 10, 12\}$ , the sequence has at least $x^{1-\delta_j}$ sign changes for any $\delta_j$ satisfying $1 - \frac{2}{(j+1)^2} < \delta_j < 1 - \frac{1}{(j+1)^2}$ .

Significance
The paper extends the understanding of the distribution of symmetric power $L$ -function coefficients beyond the standard integers to the more structured set of integers representable as sums of squares. By providing explicit asymptotic formulas for the second moments and improved bounds for the first moments across a wide range of $m$ and $j$ , the work fills gaps in the literature regarding higher symmetric powers and higher numbers of squares. The application of these results to sign change problems demonstrates that the coefficients oscillate frequently even when restricted to these sparse sequences, reinforcing the chaotic nature of these arithmetic functions. The results rely on established analytic techniques and factorization properties of $L$ -functions, contributing to the broader field of analytic number theory by refining error terms and generalizing known theorems.