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Shifted convolution sums of coefficients of symmetric power $L$ -functions with $k$ -full kernels over sums of squares in arithmetic progressions

This paper investigates the partial sums, second moments, sign changes, and shifted convolution sums of coefficients of symmetric power $L$ -functions associated with Hecke eigenforms, specifically when evaluated over sums of $m$ squares (where $m \in \{2,4,6,8,10,12\}$ ) in arithmetic progressions and twisted by $k$ -full kernel functions.

Original authors: Jewel Mahajan, Arnab Mitra

Published 2026-06-30

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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. This library doesn't contain books, but rather numbers. Specifically, it contains a special list of numbers that can be built by adding together a certain number of perfect squares (like $1^2 + 2^2$ , or $1^2 + 2^2 + 3^2$ , etc.).

The authors of this paper, Jewel Mahajan and Arnab Mitra, are acting like detectives trying to understand the hidden patterns within this list of numbers. They are looking at a specific type of mathematical "signal" attached to these numbers, which comes from something called a Symmetric Power L-function.

Here is a breakdown of their investigation using simple analogies:

1. The Characters in the Story

The Numbers (Sums of Squares): Think of these as the "addresses" in our library. The authors only care about addresses that are built by adding up 2, 4, 6, 8, 10, or 12 squares.
The Signal ( $\lambda_{sym^j f}(n)$ ): Imagine every address has a tiny, invisible sticker on it with a number written on it. Sometimes the number is positive, sometimes negative, and sometimes it's zero. This number is the "signal." The authors want to know: if we add up all these stickers for a long list of addresses, what happens? Do they cancel each other out, or do they build up a huge pile?
The Filter (Arithmetic Progression): The authors aren't looking at every address. They are only looking at addresses that leave a specific remainder when divided by a number $q$ (like looking only at houses numbered 1, 101, 201, etc.).
The "Full" Kernel: This is a special magnifying glass. The authors use a tool called a " $k$ -full kernel." Imagine this as a filter that only highlights numbers that are "very full" of prime factors (like numbers that are divisible by $p^k$ ). It helps them weigh the importance of different numbers in the list.

2. The Main Investigations

The paper tackles three main questions:

A. The "Cancellation" Test (Theorem 1 & 2)
The authors ask: "If we sum up all these signals for numbers that are sums of squares and fit our filter, do they cancel out?"

The Result: Yes, they mostly cancel out. The total sum is surprisingly small compared to the size of the list. It's like walking into a room full of people shouting, but because half are shouting "Yes" and half are shouting "No" at the exact same volume, the room sounds silent. The authors proved exactly how quiet it gets.

B. The "Energy" Test (Theorem 3 & 4)
Next, they asked: "What if we square the signals before adding them up?"

The Analogy: If the signal is a wave, squaring it is like measuring its energy. Even if the waves cancel out (positive and negative), their energy (always positive) adds up.
The Result: Here, the sum does grow. It grows in a predictable, steady way. The authors found a formula that tells you exactly how much "energy" is in the room, with a tiny bit of error margin.

C. The "Shifted" Mystery (Theorem 5 & 6)
This is the most complex part. They looked at the signal for a number $n$ , but paired it with a "kernel" (the magnifying glass) from a number $n+1$ .

The Analogy: Imagine you are comparing the sticker on House #100 with the magnifying glass on House #101. They are looking for a relationship between neighbors.
The Result: They found that even with this "shifted" comparison, the sums behave very nicely. They can predict the total sum with high precision, showing that the relationship between these neighbors follows a strict mathematical rule.

3. The "Sign Change" Detective Work (Theorem 7 & 8)

Finally, the authors asked a question about the "mood" of the signals.

The Question: Do the stickers flip back and forth between positive and negative often? Or do they stay positive (or negative) for a long time?
The Analogy: Imagine a pendulum swinging. If it swings back and forth, it's changing "sign." If it gets stuck on one side, it's not.
The Result: The authors proved that the signals are very restless. They flip from positive to negative (and back) many, many times as you go further down the list of numbers. They calculated a lower bound, proving there are at least a certain number of flips in any large section of the library. It's like proving a pendulum must swing back and forth at least 1,000 times in an hour; it can't just sit still.

Summary

In plain English, this paper is about proving that a very specific, complex mathematical pattern behaves in a predictable, balanced way.

The authors showed that:

When you add up these specific numbers, they mostly cancel each other out.
When you measure their "energy," it grows steadily.
When you look at neighbors, their relationship is stable.
The numbers flip between positive and negative very frequently, never staying in one "mood" for too long.

They achieved this by using advanced tools (like "L-functions" and "Dirichlet series") which are essentially sophisticated mathematical telescopes that allow them to see the deep structure of these numbers, even when they are filtered through complex rules like "sums of squares" and "arithmetic progressions."

Technical Summary: Shifted Convolution Sums of Coefficients of Symmetric Power L-Functions with k-Full Kernels Over Sums of Squares in Arithmetic Progressions

Problem Statement
This paper investigates the analytic behavior of partial sums and second moments of the coefficients $\lambda_{\text{sym}^j f}(n)$ associated with the $j$ -th symmetric power $L$ -function of a normalized Hecke eigenform $f$ for the full modular group $SL(2, \mathbb{Z})$ . The study focuses on sequences of integers $n$ that can be expressed as a sum of $m$ squares ( $n = a_1^2 + \dots + a_m^2$ ) and satisfy a specific congruence condition $n \equiv 1 \pmod q$ , where $q$ is a large integer and $m \in \{2, 4, 6, 8, 10, 12\}$ .

Furthermore, the authors examine the shifted convolution sums of these coefficients twisted by a $k$ -full kernel function $a(n)$ (where $k \ge 2$ ). A $k$ -full kernel function is defined such that $a(n) = a(a_2(n))$ , where $n = a_1(n)a_2(n)$ is the unique decomposition into a $k$ -free part and a $k$ -full part. The paper also analyzes the frequency of sign changes in the sequence $a(n)\lambda_{\text{sym}^j f}(n+1)$ over these specific arithmetic progressions.

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

Orthogonality of Dirichlet Characters: The sums over integers congruent to $1 \pmod q$ are transformed into sums over all Dirichlet characters modulo $q$ , separating the principal character $\chi_0$ from non-principal characters.
Factorization of Dirichlet Series: The authors construct Dirichlet series involving the coefficients $\lambda_{\text{sym}^j f}(n)$ multiplied by arithmetic functions related to the number of representations of integers as sums of squares (denoted $r_m(n)$ ). These series are factorized into products of symmetric power $L$ -functions twisted by characters and auxiliary Dirichlet series that converge absolutely in wider half-planes.
Perron's Formula and Contour Integration: To estimate the partial sums, Perron's formula is applied. The line of integration is shifted to the left (typically to $\Re(s) = 1/2 + \epsilon$ or similar), utilizing Cauchy's residue theorem to extract main terms (poles) and bounding the error terms via integrals along the new contour.
Bounds for L-Functions: The error terms are controlled using convexity bounds and mean value estimates (fourth moment bounds) for $L$ -functions, including $\zeta(s)$ , Dirichlet $L$ -functions, and symmetric power $L$ -functions twisted by characters.
Decomposition of k-Full Functions: For the shifted convolution sums involving $k$ -full kernels, the authors utilize the unique decomposition of integers into $k$ -free and $k$ -full parts. They split the summation range based on the size of the $k$ -full component and apply the previously established bounds for sums over arithmetic progressions.
Sign Change Analysis: The number of sign changes is deduced by contradiction. Assuming the sequence maintains a constant sign over a short interval, the authors compare the upper bound of the sum (derived from the shifted convolution estimates) with the lower bound of the second moment (derived from the second moment estimates). A contradiction arises if the interval is too large, implying the existence of sign changes.

Key Contributions and Results
The paper establishes asymptotic formulas and error bounds for several specific sums:

Partial Sums over Sums of Squares (Theorems 1 & 2): The authors derive bounds for $\sum \lambda_{\text{sym}^j f}(n)$ $\sum λ_{sym^{j} f} (n)$ where $n$ $n$ is a sum of $m$ $m$ squares and $n \equiv 1 \pmod q$ $n \equiv 1 (mod q)$ .
- For $m=2$ , the sum is $O(x^{1+\epsilon - \frac{1}{j+1}} q^{1+\epsilon} / \phi(q))$ .
- For $m \in \{4, 6, 8, 10, 12\}$ , the sum is $O(x^{\frac{m}{2}+\epsilon - \frac{2}{j+3}} q^{\frac{j+1}{j+3}+\epsilon} / \phi(q))$ .
Second Moments over Sums of Squares (Theorems 3 & 4): The paper provides asymptotic formulas for the second moment $\sum \lambda_{\text{sym}^j f}^2(n)$ $\sum λ_{sym^{j} f}^{2} (n)$ .
- For $m=2$ , the main term is linear in $x$ ( $c_{j,f} \frac{\phi(q)}{q} x$ ) with an error term of $O(x^{1+\epsilon - \frac{1}{(j+1)^2}} q^{1+\epsilon} / \phi(q))$ .
- For $m \in \{4, 6, 8, 10, 12\}$ , the main term is proportional to $x^{m/2}$ with an error term of $O(x^{\frac{m}{2}+\epsilon - \frac{2}{(j+1)^2}} q^{1+\epsilon} / \phi(q))$ .
Shifted Convolution Sums with k-Full Kernels (Theorems 5 & 6): The authors investigate sums of the form $\sum a(n) \lambda_{\text{sym}^j f}(n+1)$ $\sum a (n) λ_{sym^{j} f} (n + 1)$ and $\sum a^2(n) \lambda_{\text{sym}^j f}^2(n+1)$ $\sum a^{2} (n) λ_{sym^{j} f}^{2} (n + 1)$ where $n+1$ $n + 1$ is a sum of squares.
- They obtain asymptotic formulas for the second moment with a main term of order $x^{m/2}$ and explicit error bounds depending on $k$ , $j$ , and $m$ .
- For the first moment with a $k$ -full kernel, they provide upper bounds of the form $O(x^{\frac{m}{2} - \frac{2k-2}{3k(j+1)} + \epsilon})$ .
Sign Changes (Theorems 7 & 8): By combining the bounds for the first moment and the second moment, the paper proves that the sequence $a(n)\lambda_{\text{sym}^j f}(n+1)$ $a (n) λ_{sym^{j} f} (n + 1)$ changes sign frequently.
- For $m=2$ , there are at least $x^{1-\delta_j}$ sign changes in the interval $(x, 2x]$ for any $\delta_j$ in a specific range.
- For $m \in \{4, 6, 8, 10, 12\}$ , a similar lower bound on the number of sign changes is established.

Significance and Claims
The paper claims to extend previous work on shifted convolution sums (specifically those by Lü, Wang, Venkatasubbareddy, and Sankaranarayanan) by incorporating $k$ -full kernel functions and restricting the summation to integers representable as sums of $m$ squares in arithmetic progressions. The results provide explicit error terms that depend on the modulus $q$ , the number of squares $m$ , and the degree of the symmetric power $j$ . The investigation of sign changes offers new insights into the oscillatory behavior of these symmetric power coefficients when weighted by $k$ -full kernels. The authors position these results as applications of the derived bounds on partial sums and moments, without claiming broader applications beyond the scope of these specific analytic number theory problems.

1. The Characters in the Story

2. The Main Investigations

3. The "Sign Change" Detective Work (Theorem 7 & 8)

Summary

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