🔢 mathematics

Eichler-Selberg relations for the third-order mock theta functions

This paper utilizes the theory of harmonic Maass forms to establish a natural generalization of the classical Appell-Lerch series representation for Ramanujan's third-order mock theta function $f(q)$ and proves analogous Eichler-Selberg type relations for the function $\omega(q)$ .

Original authors: Wei Wang

Published 2026-06-23

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: Wei Wang

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 looking at a complex, swirling pattern of numbers that Ramanujan, a mathematical genius from the early 20th century, discovered. He called these patterns "mock theta functions." For a long time, mathematicians knew these patterns existed and behaved strangely, but they didn't have a complete map to understand exactly how they fit into the larger universe of numbers.

Think of Ramanujan's original discovery (specifically the function called $f(q)$ ) as a mysterious, half-finished puzzle. You could see the pieces, but they didn't quite lock together to form a perfect, solid picture.

The Big Breakthrough: Completing the Picture

In this paper, the author, Wei Wang, acts like a master puzzle solver who finally finds the missing pieces. He uses a modern mathematical tool called "harmonic Maass forms." You can think of this tool as a special glue that takes Ramanujan's "mock" (or incomplete) patterns and attaches them to a "non-holomorphic" (or non-perfectly smooth) partner. When you glue them together, they transform into a perfect, solid object known as a "modular form."

Once the puzzle is complete, the author can apply a powerful technique called holomorphic projection. Imagine shining a bright, pure light through this completed 3D object onto a flat wall. The shadow that appears on the wall is a new, simpler, and perfectly clean 2D pattern. This "shadow" reveals hidden relationships between the numbers that were invisible before.

The Main Discovery: A New Rule for the Numbers

The paper's main goal was to take a famous, old rule that Ramanujan and later mathematicians (like Watson) found for one specific puzzle ( $f(q)$ ) and upgrade it.

The Old Rule: It was like a simple recipe for making a specific type of cake. It worked, but it was limited to just one flavor.
The New Rule (Theorem 1): Wang has written a universal recipe book. He shows that this old rule is just one special case of a much larger family of rules. He proves that if you tweak the recipe (by changing a variable called $v$ ), you get a new, complex pattern of numbers.

Here is the surprising part: When Wang tests this new recipe for specific settings (when $v=1$ or $v=2$ ), the resulting pattern disappears completely. It vanishes into nothingness. In the world of math, when a complex pattern vanishes, it usually means a deep, hidden symmetry is at work. It's like building a complex machine only to find that, at certain speeds, it perfectly cancels itself out and becomes silent.

The Second Discovery: Another Puzzle

The paper also tackles a second, different mock theta function called $\omega(q)$ . Think of this as a sibling puzzle to the first one. It looks different, but it follows the same underlying logic. Wang proves that the same "universal recipe" works for this sibling too, revealing that they are part of the same mathematical family.

The "Eichler-Selberg" Connection

The title mentions "Eichler-Selberg relations." You can think of this as a famous, old bridge that mathematicians use to cross from one side of a river (complex, messy numbers) to the other (clean, structured numbers). Wang is building a new, wider bridge that connects these specific mock theta functions to that same river, showing us exactly how the messy numbers on one side turn into the clean numbers on the other.

Summary in Everyday Terms

The Problem: Ramanujan left behind some number patterns that were "mock" (incomplete).
The Tool: The author uses a modern method to "complete" these patterns, turning them into solid, perfect shapes.
The Action: He shines a light through these shapes to see their "shadows" (projections).
The Result: He discovers a new, general rule that explains how these numbers behave.
The Surprise: For certain settings, the result is zero, proving a beautiful symmetry.
The Legacy: This method isn't just for these two specific patterns; the author shows it's a tool that can be used to solve many other similar puzzles in the future.

The paper doesn't claim to fix cars or cure diseases; it simply says, "We found a better way to understand these specific, beautiful number patterns, and here is the new rulebook we wrote for them."

Technical Summary: Eichler-Selberg Relations for Third-Order Mock Theta Functions

Problem Statement
The paper addresses the structural properties of Ramanujan's third-order mock theta functions, specifically $f(q)$ and $\omega(q)$ . While the modularity of these functions was established through Zwegers' theory of harmonic Maass forms, the paper seeks to generalize classical identities relating these functions to Appell-Lerch series. Specifically, it aims to extend Watson's known identity for $f(q)$ , which relates a sum involving the coefficients of $f(q)$ to an Appell-Lerch series, into a broader framework of "Eichler-Selberg type" relations. These relations typically connect sums of coefficients weighted by polynomials (involving Chebyshev polynomials) to modular forms, analogous to the Eichler-Selberg class number formula.

Methodology
The author employs the theory of harmonic Maass forms and holomorphic projection to derive these relations. The methodological framework proceeds in three main stages:

Identification of Modular Components: The paper utilizes Zwegers' completion of the mock theta functions. For $f(q)$ , the function is completed to a vector-valued real-analytic modular form $H(\tau)$ of weight $1/2$ . The first component, $H_1(\tau)$ , is related to $f(q)$ via $q^{-1/24}f(q)$ plus a non-holomorphic integral involving a theta series.
Rankin-Cohen Brackets: The core technique involves constructing the Rankin-Cohen bracket $[H_1, \eta]_v$ (and similarly $[H_2, \eta]_v$ for $\omega(q)$ ), where $\eta(\tau)$ is the Dedekind eta function. This bracket combines the harmonic Maass form with a modular form to produce a real-analytic modular form of weight $2v+1$ .
Holomorphic Projection: The author applies the holomorphic projection operator $\pi_{hol}$ to these Rankin-Cohen brackets. This operator maps the real-analytic modular form to a holomorphic modular form in the space $M_{2v+1}(\Gamma_0(8), \chi)$ . The Fourier coefficients of the resulting holomorphic form are computed explicitly via an integral formula involving the non-holomorphic part of the original function.
Eisenstein Series Determination: To identify the resulting modular form, the author determines its Eisenstein component by analyzing its values at the cusps of $\Gamma_0(8)$ (specifically $i\infty, 0, 1/2, 1/4$ ). By solving a system of linear equations based on these cusp values, the Eisenstein part is expressed in terms of standard Eisenstein series, which are then converted into Lambert series.

Key Contributions and Results

Generalization of Watson's Identity (Theorem 1): The paper establishes a natural generalization of Watson's identity for $f(q)$ . For any integer $v \geq 1$ , a specific $q$ -series involving the coefficients $c_f(n)$ of $f(q)$ , weighted by Chebyshev polynomials $T_{2v}$ , is shown to be a cusp form in $S_{2v+1}(\Gamma_0(8), \left(\frac{-1}{\cdot}\right))$ .
- The identity equates a double sum involving $c_f(n)$ and $T_{2v}$ to a combination of Appell-Lerch type series and Lambert series representing Eisenstein series.
- Vanishing Results: For $v=1$ and $v=2$ , the resulting cusp form vanishes identically. This recovers known identities (including Watson's original) as special cases.
- Non-Vanishing Case: For $v=3$ , the cusp form is non-vanishing and is explicitly identified as a multiple of the eta quotient $\eta^4(z)\eta^{10}(2z)$ .
Analogous Result for $\omega(q)$ (Theorem 2): A parallel theorem is proven for the third-order mock theta function $\omega(q)$ . The paper establishes an Eichler-Selberg type relation where a $q$ -series involving coefficients $c_\omega(n)$ and Chebyshev polynomials yields a cusp form in $S_{2v+1}(\Gamma_0(8), \chi_{-4})$ .
- Similar to Theorem 1, this cusp form vanishes for $v=1$ and $v=2$ , recovering the classical identity for $\omega(q)$ , but is non-vanishing for higher weights.
Technical Lemmas: The paper provides rigorous proofs for several $q$ -series identities (Lemmas 7–11) that facilitate the transformation between Hecke-type indefinite theta series and generalized Appell-Lerch series, as well as the conversion of Eisenstein series into Lambert series.

Significance and Claims
The paper claims to provide a "natural generalization" of classical identities for third-order mock theta functions. The primary significance lies in:

Extending the Scope: It moves beyond the specific case of $v=0$ (or weight 1) to arbitrary weights $2v+1$ , revealing a family of identities parameterized by $v$ .
Methodological Extension: The author asserts that the presented method—utilizing Rankin-Cohen brackets of harmonic Maass forms and holomorphic projection—can be extended to study other classes of mock theta functions beyond the third order.
Novelty: The paper notes that the specific identities derived for $v=1$ and $v=2$ (which result in vanishing cusp forms) and the explicit form of the non-vanishing cusp form for $v=3$ have not previously appeared in the literature.

The work is framed as a contribution to the arithmetic theory of mock theta functions, linking their coefficients to the theory of modular forms and class numbers through the lens of harmonic Maass forms.