Discriminant function and paritition function - modular forms - algebra really

In summary, the student is trying to find a solution to the homework equation, but is not sure where to go next. They have found that the discriminant function is Dedekind's ##\eta-##function and Ramanujan's ##\tau-##function, but are not sure if Euler's identity holds for the special choice of ##x=q## with positive imaginary part of ##z##.
  • #1
binbagsss
1,278
11

Homework Statement



I am wanting to show that ##\Delta (t) = 1/q (\sum\limits^{\infty}_{n=0} p(n)q^{n})^{24} ##

where ##\Delta (q) = q \Pi^{\infty}_{n=1} (1-q^{n})^{24} ## is the discriminant function
and ##p(n)## is the partition function,

Homework Equations



Euler's result that : ## \sum\limits^{\infty}_{n=0} p(n)q^{n} = \Pi^{\infty}_{n=1} (1-q^{n})^{-1} ##

The Attempt at a Solution


[/B]
To be honest , I'm probably doing something really stupid, but at first sight, I would have thought we need
## \sum\limits^{\infty}_{n=0} p(n)q^{n} ## raised to a negative power, as raising to ##+24## looks like your going to get something like ##(1-q^{n})^{-24}##...

I've had a little play and get the following...

## \Delta (q) = q \Pi^{\infty}_{n=1} (1-q^{n})^{24} = \frac{q \Pi^{\infty}_{n=1}(1-q^{n})^{25}}{\Pi^{\infty}_{n=1}(1-q^{n}})=(\Pi^{\infty}_{n=1} (1-q^{n})^{25}) q \sum\limits^{\infty}_{n=1} p(n) q^{n} ##

(don't know whether it's in the right direction or where to turn next..)

Many thanks in advance.
 
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  • #2
Have you tried to use Euler's formula to write ##\Delta(q)^{-1}##?
 
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Likes binbagsss
  • #3
fresh_42 said:
Have you tried to use Euler's formula to write ##\Delta(q)^{-1}##?

erm yeah I get ##1/\Delta = 1/q (\sum\limits^{\infty}_{n=0} p(n)q^{n} ) ^{24} ## ..
 
  • #4
bump.
 
  • #5
binbagsss said:
erm yeah I get ##1/\Delta = 1/q (\sum\limits^{\infty}_{n=0} p(n)q^{n} ) ^{24} ## ..
am i missing some properties of partition function or anything?
 
  • #6
Let me list what I've found out, and what not.

We first have Dedekind's ##\eta-##function defined by ##\eta(z)=q^{\frac{1}{24}}\cdot \prod_{n \geq 1} (1-q^n)## where ##q=\exp(2 \pi i z )## and ##\mathcal{Im}(z) > 0##. This defines our modular discriminant by ##\Delta(z)=c\cdot \eta^{24}(z)## with ##c := (2\pi)^{12}##. So this results in
$$ \Delta(z) = c\cdot \eta^{24}(z) = c\cdot q \cdot \prod_{n \geq 1}(1-q^n)^{24}\; , \;\left(c := (2\pi)^{12}\; , \;q=\exp(2\pi i z)\; , \;\mathcal{Im}(z) > 0\right)$$
Then we have Ramanujan's ##\tau-##function ##\tau: \mathbb{N}\rightarrow\mathbb{Z}## defined by
$$\sum_{n=1}^{\infty}\tau(n)q^n=q\cdot \prod_{n \geq 1}(1-q^n)^{24} = \eta(z)^{24} = c^{-1} \Delta(z)$$.

Euler's identity is ##\sum_{k=0}^{\infty}p(k)x^k = \prod_{n \geq 1}(1-x^n)^{-1}## with the partition function ##p(k): \mathbb{N}\rightarrow\mathbb{N}##.

Unfortunately, I haven't found (or dug any deeper if you like) whether Euler's identity holds for our special choice ##x=q## with positive imaginary part of ##z##. Also the sums aren't over the same range ##(k \geq 0\; , \;n\geq 1)## and involve two different functions ##\tau## and ##p##. But I'm no specialist in number theory (and have only a book with basics).
 

FAQ: Discriminant function and paritition function - modular forms - algebra really

What is a discriminant function?

A discriminant function is a mathematical function that is used to classify data into different groups or categories. It takes in multiple input variables and calculates a single output value to determine the group or category that the data belongs to.

What is a partition function?

A partition function is a mathematical function that counts the number of ways that a positive integer can be expressed as a sum of smaller positive integers. It is commonly used in number theory and has applications in areas such as statistical physics and combinatorics.

What are modular forms?

Modular forms are complex-valued functions that satisfy certain transformation properties under modular transformations. They are important in number theory and have connections to many other areas of mathematics, including algebraic geometry and representation theory.

How are discriminant functions and partition functions related?

Discriminant functions and partition functions are both mathematical functions that are used in different contexts. However, they are not directly related to each other and serve different purposes. Discriminant functions are used for classification, while partition functions are used for counting and combinatorial purposes.

What is the role of algebra in discriminant functions and partition functions?

Algebra plays a crucial role in the study of discriminant functions and partition functions. Many techniques and concepts from algebra, such as group theory, ring theory, and representation theory, are used to understand and analyze these functions. Additionally, algebraic structures such as modular forms and elliptic curves are closely related to both discriminant functions and partition functions.

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