See the [website] for many specific examples.

Ramanujan's tau-function defined implicitly by

\begin{displaymath}x\prod_{n=1}^\infty (1-x^n)^{24}=\sum_{n=1}^\infty \tau(n)x^n\end{displaymath}

also yields the simplest cusp form. The associated Fourier series $\Delta(z):= \sum_{n=1}^\infty \tau(n) \exp(2 \pi i n z)$ satisfies

\begin{displaymath}\Delta\bigg(\frac{az+b}{cz+d}\bigg) =(cz+d)^{12}\Delta(z)\end{displaymath}

for all integers $a,b,c,d$ with $ad-bc=1$ which means that it is a cusp form of weight 12 for the full modular group.

The unique cusp forms of weights 16, 18, 20, 22, and 26 for the full modular group can be given explicitly in terms of (the Eisenstein series)

\begin{displaymath}E_4(z)= 1+240\sum_{n=1}^\infty \sigma_3(n) e(nz)\end{displaymath}


\begin{displaymath}E_6(z)=1-504\sum_{n=1}^\infty \sigma_5(n)e(nz)\end{displaymath}

where $\sigma_r(n)$ is the sum of the $r$th powers of the positive divisors of $n$:

\begin{displaymath}\sigma_r(n)=\sum_{d\mid n}d^r.\end{displaymath}

Then, $\Delta(z)E_4(z)$ gives the unique Hecke form of weight 16; $\Delta(z)E_6(z)$ gives the unique Hecke form of weight 18; $\Delta(z)E_4(z)^2$ is the Hecke form of weight 20; $\Delta(z)E_4(z)E_6(z)$ is the Hecke form of weight 22; and $\Delta(z)E_4(z)^2E_6(z)$ is the Hecke form of weight 26. The two Hecke forms of weight 24 are given by

\begin{displaymath}\Delta(z)E_4(z)^3 + x \Delta(z)^2\end{displaymath}

where $x=-156\pm 12\sqrt{144169}$.

An example is the L-function associated to an elliptic curve $E: y^2=x^3+Ax+B$ where $A,B$ are integers. The associated L-function, called the Hasse-Weil L-function, is

\begin{displaymath}L_E(s)= \sum_{n=1}^\infty \frac{a(n)/n^{1/2}}{n^s}=
\prod_{p\mid N}\bigg(1-\frac{a(p)/p^{1/2}}{p^s} \bigg)^{-1}\end{displaymath}

where $N$ is the conductor of the curve. The coefficients $a_n$ are constructed easily from $a_p$ for prime $p$; in turn the $a_p$ are given by $a_p=p-N_p$ where $N_p$ is the number of solutions of $E$ when considered modulo $p$. The work of Wiles and others proved that these L-functions are associated to modular forms of weight 2.

Back to the main index for The Riemann Hypothesis.