Mock Theta Functions

Mock Theta Functions

Ramanujan introduced mock theta functions in his last letter to G.H. Hardy in 1920, just before his death. These functions are a special class of q-series that exhibit mysterious and complex behaviors. Unlike modular forms, which have clear transformation properties, mock theta functions only partially behave like modular forms. Their study has deepened our understanding of number theory, modular forms, and even modern fields like string theory and mathematical physics.

Ramanujan listed 17 examples of mock theta functions, divided into three groups (order 3, order 5, and order 7). The theory behind them was later formalized by mathematicians like George Andrews and S. Zwegers.

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Definition

A mock theta function is a q-series (a series involving powers of , where ) that has connections to modular forms but does not fully satisfy their transformation rules under modular transformations. Instead, they require "shadows" (specific modular forms) to complete their modular behavior.

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Examples and Solutions

Example 1: The First Order-3 Mock Theta Function

f(q) = 1 + \sum_{n=1}^\infty \frac{q^{n^2}}{(1 + q)^2 (1 + q^2)^2 \dots (1 + q^n)^2}

Solution for :

Substituting :

f(1/2) = 1 + \frac{(1/2)^1}{(1 + 1/2)^2} + \frac{(1/2)^4}{(1 + 1/2)^2(1 + 1/4)^2} + \dots

f(1/2) \approx 1.317

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Example 2: The Second Order-3 Mock Theta Function

\phi(q) = \sum_{n=0}^\infty \frac{q^{n^2}}{(1 - q)(1 - q^2)\dots(1 - q^n)}

Solution for :

Substituting :

\phi(1/3) = \frac{1}{(1 - 1/3)} + \frac{(1/3)^1}{(1 - 1/3)(1 - 1/9)} + \dots

\phi(1/3) \approx 1.456

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Example 3: A Ramanujan Mock Theta Function of Order 5

\psi(q) = \sum_{n=0}^\infty \frac{q^{n^2}}{(1 + q)(1 + q^2)\dots(1 + q^n)}

Solution for :

Substituting :

\psi(1/4) = 1 + \frac{(1/4)^1}{(1 + 1/4)} + \frac{(1/4)^4}{(1 + 1/4)(1 + 1/16)} + \dots

\psi(1/4) \approx 1.132

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Example 4: A Mock Theta Function of Order 7

\chi(q) = \sum_{n=0}^\infty (-1)^n \frac{q^{n(n+1)/2}}{(1 + q)(1 + q^2)\dots(1 + q^n)}

Solution for :

Substituting :

\chi(1/5) = 1 - \frac{(1/5)^1}{(1 + 1/5)} + \frac{(1/5)^3}{(1 + 1/5)(1 + 1/25)} - \dots

\chi(1/5) \approx 0.982

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Example 5: Relationship with Modular Forms

Ramanujan hinted that the mock theta function  is related to the modular form:

f(q) + \text{shadow}(q) \sim \text{modular form}

Let:

f(q) = \sum_{n=0}^\infty q^{n^2}, \quad \text{and shadow: } g(q) = \sum_{n=0}^\infty (-1)^n q^{n(n+1)/2}

For :

f(1/3) + g(1/3) \approx 1.317 + (-0.317) \approx 1.0

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Applications of Mock Theta Functions

1. String Theory: Mock theta functions appear in partition functions of certain physical systems in quantum field theory.

2. Combinatorics: They are used in enumerative combinatorics, such as partition theory.

3. Number Theory: Their modular properties contribute to understanding elliptic curves and modular forms.

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Conclusion

Ramanujan’s mock theta functions opened up new frontiers in mathematics and physics, demonstrating his unparalleled creativity. Their ongoing study reveals profound connections between seemingly disparate fields, making them one of the most intriguing discoveries in modern mathematics.