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On the discrete mean square of certain hybrid sum involving $a_{\mathbb{K}}(n)$

This paper establishes an asymptotic formula with a tight error term for the discrete mean square of the hybrid sum involving the squared coefficients $a_{\mathbb{K}}^2(n)$ , where $n$ is represented as a sum of eight squares, for a non-normal cubic algebraic number field $\mathbb{K}$ with negative discriminant.

Original authors: Ekta Soni, M. S. Datt, A. Sankaranarayanan

Published 2026-03-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ekta Soni, M. S. Datt, A. Sankaranarayanan

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

The Big Picture: Counting Hidden Patterns in a Cosmic Grid

Imagine you have a giant, invisible 8-dimensional grid. In this grid, every point is defined by 8 numbers (like coordinates on a map, but with 8 directions instead of 2).

The mathematicians in this paper are trying to solve a massive counting puzzle. They want to know: If we look at all the points in this grid that are within a certain distance from the center, how much "weight" do they carry?

But here's the twist: The "weight" of each point isn't just 1. It's determined by a very complex, hidden rule based on a specific type of number system called a Cubic Number Field.

The Characters in Our Story

The Grid (The Hybrid Sum):
Think of the grid as a giant dance floor. The dancers are points $(n_1, n_2, ..., n_8)$ . The rule is that the sum of their squares ( $n_1^2 + ... + n_8^2$ ) must be less than a giant number $x$ . This is like saying, "Only dancers within a certain radius of the center are allowed to dance."
The Hidden Rule (The Dedekind Zeta Function):
Every dancer has a secret score attached to them, called $a_K(n)$ . This score comes from a "number field" (a special universe of numbers).
- Analogy: Imagine a factory that produces numbers. Sometimes it makes 1, sometimes 2, sometimes 100. The rule for how many items it makes depends on the "shape" of the factory (the cubic polynomial). The paper studies the square of these scores ( $a_K(n)^2$ ). Squaring them emphasizes the big scores and ignores the tiny ones, giving us a measure of the "energy" in the system.
The Goal (The Asymptotic Formula):
The authors want to add up the scores of all the dancers inside the circle.
- The Problem: Adding up millions of these scores one by one is impossible. It's like trying to count every grain of sand on a beach by picking them up individually.
- The Solution: They want a formula that predicts the total sum based only on the size of the circle ( $x$ ), without counting every single grain.

The Journey: How They Solved It

The paper is a detective story where the authors use advanced tools to find a shortcut.

Step 1: Breaking Down the Mystery (Preliminaries)

The authors first realize that the "score" ( $a_K(n)$ ) is related to something called Hecke Eigenforms.

Analogy: Think of a musical instrument (like a violin). When you play a note, it vibrates in a specific pattern. These patterns are the "eigenforms." The authors use the mathematics of these vibrations to understand the behavior of the number scores. They also use Dirichlet characters, which are like filters that sort numbers into different categories (like sorting red marbles from blue marbles).

Step 2: The Magic Lens (The L-Function)

To count the grains of sand without picking them up, the authors use a "Magic Lens" called a Dirichlet Series (specifically, an L-function).

Analogy: Imagine you want to know the total volume of water in a river. Instead of measuring every drop, you look at the river's flow rate over time. The L-function is like a super-spectrometer that takes the chaotic, jagged data of the individual numbers and turns it into a smooth, flowing curve.
They prove that this curve has a specific shape with a "pole" (a spike) at a specific point. This spike is the key.

Step 3: The Residue Theorem (The Shortcut)

The authors use a powerful mathematical tool called Cauchy's Residue Theorem.

Analogy: Imagine you are trying to find the treasure buried in a field. Instead of digging everywhere, you use a metal detector. The detector beeps loudly at one specific spot (the pole). The authors realize that the entire sum of the dancers is mostly determined by the strength of that beep.
The "Main Term" ( $Cx^4 P_1(\log x)$ ) is the result of this beep. It tells them the sum grows roughly like $x^4$ (the size of the grid to the power of 4), with a little extra wiggle room from the logarithm.

Step 4: Cleaning Up the Noise (The Error Term)

The formula isn't perfect. There is always some "noise" or "error" because the river isn't perfectly smooth.

The Challenge: They need to prove that this noise is small enough to be ignored.
The Method: They use a technique called Perron's Formula, which is like drawing a box around the river. They calculate the flow inside the box and then estimate how much water might leak out the sides.
The Result: They prove the noise is roughly $x^{198/53}$ $x^{198/53}$ .
- Why this matters: $198/53$ is approximately $3.73$. Since the main answer is $x^4$ , the error ( $x^{3.73}$ ) is significantly smaller than the answer. It's like saying, "The total weight of the beach is 1 billion tons, give or take 10 million tons." That's a very tight, accurate prediction!

The Takeaway

In simple terms, this paper does the following:

Identifies a complex pattern in a specific type of number system.
Uses the "music" of numbers (modular forms) to translate a hard counting problem into a smooth curve.
Calculates the total sum of these numbers for a massive range.
Provides a precise prediction (an asymptotic formula) that tells us exactly how this sum grows as the range gets larger, with a very small margin of error.

The Bottom Line: The authors successfully built a mathematical "crystal ball" that predicts the behavior of a chaotic, high-dimensional sum of numbers with surprising accuracy, proving that even in the most complex number systems, there is an underlying order that can be measured.

1. Problem Statement

The paper investigates the asymptotic behavior of the second-order moment (mean square) of the coefficients $a_K(n)$ of the Dedekind zeta function $\zeta_K(s)$ , summed over a specific hybrid sequence of natural numbers.

Context: Let $K$ be a non-normal algebraic number field of cubic degree over $\mathbb{Q}$ , defined by an irreducible polynomial $x^3 + ax^2 + bx + c$ with a negative discriminant ( $D_K < 0$ ).
Coefficients: $a_K(n)$ denotes the number of integral ideals in the ring of integers $\mathcal{O}_K$ with norm $n$ .
The Hybrid Sum: The authors study the sum:
$S(x) = \sum_{\substack{n = \sum_{i=1}^8 n_i^2 \le x \\ (n_1, \dots, n_8) \in \mathbb{Z}^8}} a_K^2(n)$
This represents a "hybrid" sum because it combines the arithmetic properties of the number field $K$ (via $a_K(n)$ ) with the geometric properties of the sum of eight squares (via the condition $n = \sum n_i^2$ ).
Goal: To establish an asymptotic formula for $S(x)$ as $x \to \infty$ with a tight error term.

2. Methodology

The proof employs a combination of analytic number theory techniques, specifically focusing on Dirichlet series, modular forms, and contour integration.

A. Decomposition of Coefficients

The authors utilize the relationship between the Dedekind zeta function of a cubic non-normal field and modular forms.

They express $\zeta_K(s)$ as $\zeta(s)L(s, f)$ , where $f$ is a holomorphic cusp form of weight 1 for the congruence subgroup $\Gamma_0(|D_K|)$ .
Consequently, the coefficients are given by the Dirichlet convolution: $a_K(n) = \sum_{d|n} \lambda_f(d)$ , where $\lambda_f(n)$ are the normalized Hecke eigenvalues of $f$ .

B. Analysis of the Sum of Squares

The inner sum over the condition $n = \sum_{i=1}^8 n_i^2$ is handled using the representation function $r_8(n)$ , which counts the number of ways to write $n$ as a sum of 8 squares.

Using known identities, $r_8(n) = 16(-1)^n \sum_{d|n} (-1)^d d^3$ .
The authors define an auxiliary arithmetic function $g(n) = (-1)^n \sum_{d|n} (-1)^d d^3$ and prove (Lemma 2.1) that $g(n)$ is multiplicative.
The original sum is transformed into a sum over $n \le x$ involving the product $a_K^2(n)g(n)$ .

C. Construction of the Dirichlet Series

The core of the method involves constructing the Dirichlet series $F(s)$ associated with the coefficients $a_K^2(n)r_8(n)$ :
$F(s) = \sum_{n=1}^\infty \frac{a_K^2(n)r_8(n)}{n^s}$

Factorization: The authors factorize $F(s)$ into a product of standard $L$ -functions. They show that:
$F(s) = 16 \frac{B_2(s)}{A_2(s)} \zeta^2(s-3) L^2(s-3, f) L(s-3, \text{sym}^2 f) \zeta^2(s) L^2(s, f) L(s, \text{sym}^2 f) B(s)$
where $B(s)$ is an absolutely convergent "harmless" factor, and $A_2(s), B_2(s)$ are factors related to the prime 2.
Pole Analysis: The term $\zeta^2(s-3)$ introduces a pole of order 2 at $s=4$ . This dictates the main term of the asymptotic formula.

D. Contour Integration (Perron's Formula)

To recover the partial sum from the Dirichlet series, the authors apply Perron's formula:
$\sum_{n \le x} a_K^2(n)r_8(n) = \frac{1}{2\pi i} \int_{c-iT}^{c+iT} F(s) \frac{x^s}{s} ds$

Contour Shift: The line of integration is shifted from $\text{Re}(s) = 4+\epsilon$ to $\text{Re}(s) = 7/2 + \epsilon$ .
Residue Theorem: By Cauchy's residue theorem, the integral equals the residue at the pole $s=4$ (the main term) plus the integrals along the new vertical line and the horizontal segments.

E. Error Term Estimation

The error term is derived by bounding the integrals along the shifted contour.

Bounds Used: The authors utilize convexity bounds and mean value estimates for the Riemann zeta function $\zeta(s)$ , the $L$ -function $L(s, f)$ , and the symmetric square $L$ -function $L(s, \text{sym}^2 f)$ .
Key Lemmas: They apply specific bounds (Lemmas 2.3–2.8) regarding the growth of these functions in the critical strip and on the line $\text{Re}(s) = 1/2 + \epsilon$ .
Optimization: The parameter $T$ (height of the contour) is optimized to balance the error terms arising from the horizontal and vertical integrals.

3. Key Contributions

Extension of Moment Problems: While previous works (Landau, Chandrasekharan, Good, Naveen & Prashant) studied moments of $a_K(n)$ over standard intervals or specific sequences, this paper addresses a hybrid sum involving the sum of eight squares, a more complex arithmetic structure.
Factorization of Hybrid L-functions: The paper successfully decomposes the Dirichlet series associated with the hybrid sum $a_K^2(n)r_8(n)$ into a product involving $\zeta(s)$ , $L(s, f)$ , and $L(s, \text{sym}^2 f)$ . This requires careful handling of the multiplicative properties of $r_8(n)$ and the specific structure of the cubic field.
Tight Error Term: The authors achieve a significantly tight error term compared to trivial bounds, demonstrating the efficacy of combining modular form theory with classical analytic techniques.

4. Main Result

The paper proves the following Main Theorem:

For sufficiently large $x$ and any $\epsilon > 0$ :
$\sum_{\substack{n = \sum_{i=1}^8 n_i^2 \le x \\ (n_1, \dots, n_8) \in \mathbb{Z}^8}} a_K^2(n) = C x^4 P_1(\log x) + O\left(x^{\frac{198}{53} + \epsilon}\right)$

Main Term: $C x^4 P_1(\log x)$ , where $C$ is a constant and $P_1(\log x)$ is a polynomial of degree 1 in $\log x$ . The degree 4 in $x$ corresponds to the pole of order 2 at $s=4$ (since the residue of $x^s/s$ at a double pole yields $x^4 \log x$ ).
Error Term: $O(x^{\frac{198}{53} + \epsilon})$ $O (x^{\frac{198}{53} + ϵ})$ .
- Note: $\frac{198}{53} \approx 3.7358$ . This is a significant improvement over the trivial bound of $x^4$ and represents a precise estimation of the fluctuation.
- The exponent is derived by choosing $T = x^{14/53}$ to balance the terms $x^4/T$ and $x^{7/2}T^{25/28}$ .

5. Significance

Analytic Number Theory: The work contributes to the understanding of the distribution of ideal counts in non-normal cubic fields, a notoriously difficult area due to the lack of a simple Euler product structure for $\zeta_K(s)$ compared to normal extensions.
Hybrid Sums: It provides a template for analyzing sums where arithmetic coefficients are weighted by geometric constraints (like sums of squares), bridging the gap between algebraic number theory and the theory of quadratic forms.
Methodological Rigor: The rigorous handling of the "harmless" factors at the prime 2 and the precise application of bounds for symmetric square $L$ -functions demonstrate advanced techniques in estimating mean values of $L$ -functions.

In summary, the paper establishes a precise asymptotic formula for a complex hybrid sum, linking the arithmetic of cubic fields to the geometry of 8-dimensional spheres, with a highly optimized error term derived through sophisticated contour integration and $L$ -function bounds.