Notes on certain binomial harmonic sums of Sun's type

This paper proves and generalizes Z.-W. Sun's conjectures on infinite series involving harmonic numbers and binomial coefficients by evaluating them in closed form through their interpretation as automorphic objects on the moduli spaces of Legendre curves with genera 1, 2, 3, and 5.

The Gist

Imagine you are a detective trying to solve a massive, infinite puzzle. The pieces of this puzzle are numbers that keep adding up forever (infinite series). Some of these numbers are simple, but others are "harmonic"—they involve sums of fractions like $1 + 1/2 + 1/3 + \dots$—and they are mixed with "binomial coefficients," which are the numbers you get when you count combinations (like how many ways you can pick a team from a group).

For years, a mathematician named Zhi-Wei Sun has been finding these puzzles and guessing the final answers. He has a notebook full of these guesses (conjectures), but he hasn't always been able to prove why they are true. He's like someone who sees a pattern in the clouds and says, "I bet that cloud looks like a dragon," but hasn't yet built the dragon to prove it.

Yajun Zhou, the author of this paper, is the master builder who finally constructs the dragons.

Here is how the paper works, explained through simple analogies:

1. The Problem: The "Sun's Series" Puzzle

Sun's puzzles look like this:
$$\text{Sum of } \left( \text{Combinations} \times \text{Harmonic Numbers} \times \text{Powers} \right)$$
These sums are notoriously difficult. They are like trying to balance a stack of Jenga blocks where every block is slightly different, and the tower goes up to infinity. Sun has guessed the height of the tower for many specific cases, but proving it requires a very special kind of ladder.

2. The Tool: The "Legendre Function" Ladder

Zhou's main tool is something called a Legendre Function.

• The Analogy: Imagine a magical, shape-shifting bridge. In mathematics, this bridge connects two different worlds:
  1. The World of Discrete Numbers: The messy, step-by-step world of infinite sums (the Jenga tower).
  2. The World of Continuous Geometry: The smooth, flowing world of curves and shapes (specifically, shapes called "Legendre curves").

Zhou realizes that if you can translate the messy sum into a smooth curve, you can use the laws of physics and geometry to solve it. He treats these sums not just as numbers, but as automorphic objects—think of them as patterns that repeat perfectly on a curved surface, like a wallpaper that wraps around a sphere without any seams.

3. The Method: "Tweaking" the Bridge

The paper uses a technique called differentiation (or "tweaking").

• The Analogy: Imagine you have a machine that produces a specific sound (a sum). If you slightly adjust a dial on the machine (changing a parameter by a tiny amount, $\epsilon$), the sound changes.
• Zhou takes the "dial" and turns it just a hair. This tiny turn introduces the "Harmonic Numbers" (the fractions) into the equation. By listening to how the sound changes when he tweaks the dial, he can decode the exact value of the original sum. It's like figuring out the weight of a hidden object by seeing how much a spring stretches when you add a tiny pebble to it.

4. The "Modular" Map

The paper relies heavily on Modular Forms.

• The Analogy: Think of a map of a city. A normal map shows streets. A modular map is like a magical map that shows the city from every angle at once. If you rotate the map, the city looks the same, but the numbers change in a predictable way.
• Zhou uses these maps to find "special points" (called CM points). At these specific points, the messy infinite sums suddenly collapse into clean, simple answers involving famous constants like $\pi$ (pi), $\Gamma$ (gamma functions), and logarithms. It's like finding a secret shortcut that bypasses the traffic jam of infinite calculation.

5. The Results: Solving Sun's Conjectures

The paper does three main things:

1. Proves Sun's Guesses: It confirms that Sun's guesses about these infinite sums were correct. For example, it proves that a specific sum involving combinations and harmonic numbers equals a specific combination of $\pi$ and $\sqrt{2}$.
2. Generalizes the Rules: It doesn't just solve Sun's specific cases; it creates a "master key" that can unlock many similar puzzles that Sun didn't even guess yet.
3. Connects to Physics: The paper mentions that these mathematical structures appear in Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD).
   • The Analogy: These infinite sums aren't just abstract math games; they are the "accounting books" of the universe. When physicists calculate how particles interact, they end up with these exact same messy sums. Zhou's work helps physicists calculate these interactions more accurately.

Summary

In simple terms, Yajun Zhou took a collection of difficult, infinite number puzzles proposed by Zhi-Wei Sun. He built a bridge using Legendre functions (smooth curves) and modular maps (geometric patterns) to translate these messy sums into smooth geometry. By "tweaking" these geometric shapes, he was able to prove Sun's guesses and discover new, beautiful formulas that connect the world of pure number theory to the fundamental laws of physics.

He didn't just solve the puzzle; he showed us that the puzzle pieces were actually parts of a much larger, beautiful picture all along.

Technical Summary

Here is a detailed technical summary of the paper "Notes on Certain Binomial Harmonic Sums of Sun's Type" by Yajun Zhou.

1. Problem Statement

The paper addresses a class of infinite series involving products of binomial coefficients and harmonic numbers, often referred to as "binomial harmonic sums." These sums were recently conjectured by Z.-W. Sun in a series of papers (notably [80, 81]). The general form of these series is:
$$\sum_{k=1}^{\infty} \binom{2k}{k}^N \frac{1}{k^s} \left( \frac{T}{2^{2N}} \right)^k \left[ \prod H^{(r_j)}_{k-1} \right] \left[ \prod H^{(r'_j)}_{2k-1} \right]$$
where $N, s \in \mathbb{Z}$, $M, M' \in \mathbb{Z}_{\geq 0}$, and $|T| < 1$.

While previous works had evaluated cases where $N \in \{-1, 0, 1\}$ using motivic periods and Goncharov's polylogarithms, the specific challenge tackled in this paper is the evaluation of these series for $N \in \{2, 3, 4\}$. These cases correspond to higher-degree binomial coefficients (e.g., $\binom{2k}{k}^2$, $\binom{2k}{k}\binom{3k}{k}$, $\binom{2k}{k}^2\binom{4k}{2k}$) and are deeply connected to the arithmetic of Legendre curves of positive genus ($g \in \{1, 2, 3, 5\}$).

2. Methodology

The author employs a sophisticated blend of analytic number theory, modular forms, and special functions. The core methodology involves:

• Legendre Functions and Hypergeometric Series: The series are interpreted as perturbations (specifically $\varepsilon$-expansions) of Legendre functions $P_\nu(1-2t)$ and generalized hypergeometric series $_pF_q$. By differentiating these functions with respect to their degree $\nu$ or parameters, the author generates the harmonic number terms ($H_k, H_k^{(2)}$, etc.) naturally appearing in the summands.
• Modular Parametrization: The sums are linked to modular invariants (such as the $j$-invariant and Ramanujan's $\alpha_N$ functions) and CM points (Complex Multiplication points) on the upper half-plane. This allows the evaluation of the series at specific algebraic arguments to be reduced to closed forms involving Gamma functions, logarithms, and $\pi$.
• Clausen Couplings: The paper utilizes Clausen's formula to represent products of Legendre functions as generalized hypergeometric series. This is crucial for handling cubic and quartic binomial terms (e.g., $\binom{2k}{k}^3$).
• Automorphic Green's Functions and Epstein Zeta Functions: For series involving derivatives with respect to the degree $\nu$, the author connects the sums to automorphic Green's functions of weight 4 and Epstein zeta functions on congruence subgroups $\Gamma_0(N)$.
• Clebsch–Gordan Couplings: The paper reinterprets integrals over products of three Legendre functions (Clebsch–Gordan integrals) to derive new series identities, utilizing Bailey's reduction formulas and Meijer G-functions.

3. Key Contributions and Results

A. Generalization of Sun's Conjectures (Sections 2 & 3)

• Legendre–Sun Series: The author proves and generalizes Sun's conjectures for $N=2,3,4,6$ (corresponding to $\nu \in \{-1/2, -1/3, -1/4, -1/6\}$).
  • Theorem 2.1: Establishes closed-form identities for series like $\sum \binom{2k}{k}^2 (t/24)^k (2H_{2k} - H_k)$ in terms of $P_\nu(1-2t)$ and logarithms.
  • Modular Reduction: When the argument $t$ corresponds to a CM point, these series evaluate to explicit algebraic combinations of $\log(\text{algebraic numbers})$, $\pi$, and values of the Gamma function at rational arguments (e.g., $\Gamma(1/4), \Gamma(1/3)$).
• Ramanujan–Sun Series: The paper provides a detailed proof of Sun's Conjecture 29 (related to the Chudnovsky series for $1/\pi$) using Clausen couplings.
  • Theorem 3.1: Derives a family of series identities involving $\binom{2k}{k}^3$ and harmonic numbers, linking them to the imaginary parts of ratios of Legendre functions.
  • Explicit Evaluation: Confirms the specific series for $1/\pi$involving the constant$X_{163} = -1/(2^{18} \cdot 3^3 \cdot 5^3 \cdot 23^3 \cdot 29^3)$ and its companion series with harmonic numbers.

B. Automorphic Green's Functions and Epstein Zeta Functions (Section 4)

• Green–Sun Series: The author introduces series representations for automorphic Green's functions of weight 4.
  • Theorem 4.1: Expresses infinite sums involving $\psi$-functions (digamma) as integrals of Legendre functions.
  • Epstein–Sun Series: Theorem 4.2 connects series involving second-order harmonic numbers ($H_k^{(2)}$) to Epstein zeta functions $E_{\Gamma_0(4)}(z, 2)$.
  • Arithmetic Significance: Corollary 4.6 proves that for CM points, the exponential of certain linear combinations of these series yields algebraic numbers, confirming a conjecture related to Gross–Zagier heights.

C. Clebsch–Gordan Couplings (Section 5)

• New Perspective on Integrals: The paper revisits integrals of the form $\int_{-1}^1 P_\mu(x) P_\nu(x) P_\nu(-x) dx$.
• Theorem 5.1: Establishes a correspondence between these integrals and derivatives of generalized hypergeometric series $_4F_3$.
• New Evaluations: This approach yields closed forms for previously difficult sums, such as:
  $$\sum_{k=0}^\infty \frac{\binom{2k}{k}^4}{28^k (4k+1)} = \frac{\Gamma(1/4)^8}{96\pi^5}$$
  and sums involving $H_k^{(2)}$ and $H_k^{(4)}$ that were not covered by Sun's original conjectures.

4. Significance

• Resolution of Conjectures: The paper provides rigorous analytic proofs for numerous experimental conjectures by Z.-W. Sun, moving them from numerical observation to mathematical certainty.
• Unification of Fields: It demonstrates a deep connection between combinatorial series (binomial harmonic sums), elliptic curves (Legendre curves), and automorphic forms (Green's functions, Epstein zeta functions).
• New Analytic Tools: The introduction of "Green–Sun" and "Epstein–Sun" series expands the toolkit for evaluating infinite sums, particularly those involving higher powers of central binomial coefficients ($N \geq 2$).
• Arithmetic Geometry: The results provide concrete examples of the arithmetic properties of automorphic Green's functions at CM points, supporting the broader Gross–Zagier conjectures regarding the algebraicity of these values.

In summary, Yajun Zhou's work successfully generalizes Sun's binomial harmonic sums by interpreting them as automorphic objects on moduli spaces of Legendre curves, thereby providing closed-form evaluations for a wide class of previously intractable infinite series.