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The Period-Three Secondary Term in the Mean Value of $L(1/2,\chi_D)$ in the Hyperelliptic Ensemble

This paper corrects and generalizes Florea's asymptotic formula for the first moment of quadratic Dirichlet $L$ -functions over the odd-degree hyperelliptic ensemble by identifying a previously overlooked period-three secondary term arising from nonreal poles and extending the result to all odd prime powers.

Original authors: Hwanyup Jung

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hwanyup Jung

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

In the vast landscape of number theory, mathematicians often study patterns hidden within the distribution of prime numbers. To make these patterns easier to see, they sometimes move their investigation from the familiar world of whole numbers to a different mathematical universe built from polynomials over finite fields. Imagine a world where numbers are not infinite strings of digits, but rather expressions made from variables and coefficients, where the "primes" are specific types of irreducible polynomials. In this setting, researchers can ask questions about how certain mathematical functions behave on average. One such question concerns the central values of quadratic Dirichlet L-functions, which are complex functions that encode deep information about these polynomial primes. Specifically, mathematicians look at the average value of these functions across a large collection of polynomials, known as the hyperelliptic ensemble. Understanding this average helps reveal the underlying structure of the number system, much like counting the average height of trees in a forest reveals the health of the ecosystem.

For some time, the accepted answer to this question included a main prediction and a secondary correction term. This secondary term was thought to be a single, smooth mathematical curve that grew steadily as the size of the polynomials increased. It was derived from a specific feature in the complex equations used to calculate the average, a feature that corresponded to a single point on the real number line. The prevailing view was that this single point provided the complete picture of the secondary behavior, with any remaining discrepancies being small enough to ignore. However, a recent investigation by Hwanyup Jung has revealed that this picture was incomplete. By revisiting the calculation with a more careful approach that works for all types of finite fields, not just the special cases previously studied, Jung discovered that the secondary term is not a single smooth curve at all. Instead, it is a more complex phenomenon that splits into three distinct trends, depending on a simple property of the size of the polynomials.

The core of Jung's discovery lies in the geometry of the equations used to solve the problem. When mathematicians analyze these averages, they often transform the problem into a search for specific points, called poles, in a complex plane. These poles act like anchors that determine the shape of the final answer. In the previous calculation, researchers focused on one prominent pole located on the positive real axis. This pole produced the single smooth curve that had been accepted as the secondary term. Jung's work showed that this was only half the story. The equations actually possess three poles of equal importance, all sitting at the same distance from the center of the mathematical plane. One of these is the real pole that everyone had seen, but the other two are complex conjugates, existing as a pair of non-real numbers that mirror each other.

Because these three poles are equally strong, they all contribute to the final answer with the same level of significance. The contribution from the real pole matches the old, smooth curve. However, the two complex poles introduce a new element: a rhythmic oscillation that depends on the remainder when the size of the polynomial is divided by three. This means the average value does not follow a single line. Instead, it splits into three separate lines, each corresponding to a different remainder. If the size of the polynomial leaves a remainder of zero, the average follows one path; if it leaves a remainder of one, it follows a second; and if it leaves a remainder of two, it follows a third. These three paths are not just slightly different; they are distinct linear trends that can diverge significantly from one another. In some cases, one of these paths might even slope in the opposite direction compared to the average of the three.

This finding overturns the previous assumption that the secondary term was a single, unified expression. Jung proved that the old formula was actually just the average of these three distinct trends, a value that no single case in the real world actually follows. The paper demonstrates that the behavior of these mathematical functions is governed by a period-three phenomenon, a cycle that repeats every time the size of the polynomial increases by a certain amount. To confirm this, the author performed exact calculations for a specific case where the field size is three. The results showed the data points clearly separating into three distinct groups, each tracking its own unique line, exactly as the new theory predicted. This separation was so pronounced that it could not be explained away as a minor error or a temporary fluctuation.

The work also resolves a technical limitation in previous research. Earlier studies had assumed that the size of the field had to be a prime number with a specific property to make the calculations work. Jung introduced a refined method that removes this restriction, allowing the result to hold for any odd prime power. This was achieved by adjusting the mathematical tools used to handle the phases of the numbers, ensuring that the multiplicative properties required for the proof remained valid in all cases. By doing so, the paper establishes a complete and rigorous formula for the average value, capturing the full complexity of the secondary term. The result is a more accurate and nuanced understanding of how these mathematical functions behave, revealing that what looked like a single, steady trend was actually a trio of distinct, interwoven patterns, each waiting to be seen when the right lens is applied.

Technical Summary of "The Period-Three Secondary Term in the Mean Value of $L(1/2, \chi_D)$ in the Hyperelliptic Ensemble"

Problem Statement
The paper investigates the first moment of quadratic Dirichlet $L$ -functions at the central point over the odd-degree hyperelliptic ensemble $\mathcal{H}_{2g+1}$ in the function field $\mathbb{F}_q[T]$ , where $q$ is an odd prime power. The quantity of interest is:
$M_1(g) := \sum_{D \in \mathcal{H}_{2g+1}} L\left(\frac{1}{2}, \chi_D\right)$
as the genus $g \to \infty$ with $q$ fixed. While Andrade and Keating determined the principal term, Florea previously established a secondary term of size $g q^{2g/3}$ . Florea's result, however, relied on the assumption that $q \equiv 1 \pmod 4$ and identified the secondary term as a single polynomial in $g$ derived from the positive real pole of a generating function. This paper challenges the completeness of Florea's secondary term and extends the validity of the asymptotic formula to all odd prime powers $q$ .

Methodology
The author employs the square-free sieve and Poisson summation to decompose the moment into zero-frequency, square-dual, and nonsquare-dual contributions. Key methodological innovations include:

Phase-Normalized Gauss Sums: To handle arbitrary odd prime powers $q$ (specifically the case $q \equiv 3 \pmod 4$ ), the author introduces a phase-normalized quadratic Gauss sum. This normalization restores the multiplicativity of the Gauss sums and the validity of the local Euler product evaluations used in the square-dual calculation, which fail for unnormalized sums when $q \equiv 3 \pmod 4$ .
Contour Integration and Residue Analysis: The square-dual contribution is evaluated via a double contour integral. The author deforms the contour to a "secondary circle" $|z| = q^{-4/3}$ . Unlike the analysis in [3], which considers only the positive real pole on this circle, this paper identifies that the denominator $(1 - q^4 z^3)^2$ creates three double poles of equal modulus: $z_k = q^{-4/3} \omega^k$ for $k=0, 1, 2$ , where $\omega = e^{2\pi i/3}$ .
Residue Bookkeeping: The author calculates the residues at all three poles simultaneously. The contributions from the complex conjugate poles ( $z_1, z_2$ ) are shown to be of the same order as the real pole ( $z_0$ ) and depend on the genus $g$ modulo 3.

Key Contributions and Results

Correction of the Secondary Term: The paper demonstrates that Florea's secondary term corresponds only to the contribution of the positive real pole ( $z_0$ ). The complete secondary term includes contributions from the two nonreal conjugate poles ( $z_1, z_2$ ).
Period-Three Phenomenon: The complete secondary term is shown to be:
$q^{2g/3} \left( \alpha_{g \bmod 3} g + \beta_{g \bmod 3} \right)$
where the coefficients $\alpha_m, \beta_m$ depend on $q$ and exhibit a minimal period of exactly 3 with respect to the genus $g$ . The dependence is genuine; the coefficients for different residue classes are not merely small perturbations of a single average. For $q=3$ , the coefficients vary significantly, with one class having an opposite sign to the mean.
Generalization to All Odd Prime Powers: Theorem 1.1 is proved for every odd prime power $q$ . This resolves the gap in [3], which left the case $q \equiv 3 \pmod 4$ untreated due to the failure of multiplicativity in unnormalized Gauss sums. The phase-normalized Gauss sum allows the argument to proceed uniformly.
Exact Finite-Ensemble Verification: For $q=3$ , the author computes exact moments for genera up to $g=15$ . The numerical data confirms the theoretical prediction: when the data is separated by $g \bmod 3$ , the points align with three distinct linear trends rather than a single line with oscillatory error.

Significance and Claims
The paper claims that the secondary term in the mean value of these $L$ -functions is not a single polynomial in $g$ , but a periodic function of $g$ with period 3. The "period-three phenomenon" arises because the secondary circle in the complex plane carries three double poles of equal modulus, all of which must be retained in the asymptotic expansion.

The author asserts that the result is a completion of Florea's calculation rather than a re-normalization. Florea's polynomial is identified as the average of the three period-dependent terms over the residue classes. The nonreal poles contribute a term of order $g q^{2g/3}$ on at least two of the three residue classes, a magnitude that exceeds the error term $O(q^{g(1+\epsilon)/2})$ and cannot be absorbed into it.

The paper does not claim to alter the principal term (determined by Andrade and Keating) nor does it propose new applications beyond the refinement of the asymptotic formula for the first moment in the hyperelliptic ensemble. The primary contribution is the rigorous identification of the periodic secondary term and the extension of the proof to all odd prime powers via phase normalization.

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