Lagmental Vicfred

Eisenstein Series Turn Divisor Sums into Fourier Coefficients by Vicfred

tags: mathematics, modular forms, q-series, partitions

The normalized Eisenstein series of weights four and six have coefficients given by sigma three and sigma five. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

Notation

A modular form \(f\) on the upper half-plane transforms predictably under fractional linear maps and has a Fourier expansion in \(q=e^{2\pi i\tau}\). Its coefficients often encode arithmetic.

$$ E_4(\tau)=1+240\sum_{n\ge1}\sigma_3(n)q^n $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ E_6(\tau)=1-504\sum_{n\ge1}\sigma_5(n)q^n $$

Stress the formula

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \begin{aligned}E_4&=1+240q+2160q^2+6720q^3+\cdots,\\E_6&=1-504q-16632q^2-122976q^3+\cdots.\end{aligned} $$

Interpretation

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad E_4(\tau)=1+240\sum_{n\ge1}\sigma_3(n)q^n,\\[5pt] \mathsf{C}\;&:\quad E_6(\tau)=1-504\sum_{n\ge1}\sigma_5(n)q^n. \end{aligned} $$

Limit of the argument

Weight, level, character, and cusp conditions are part of the definition. A formal \(q\)-series is not automatically a modular form.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] E_6(\tau)=1-504\sum_{n\ge1}\sigma_5(n)q^n \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Mon 07 July 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.