Celebrating Ramanujan's birthday — From powers of divisors to coefficients of modular forms

In a Twitter post, Anton Hilado reminded us that today (December 22nd) was the birthday of Srinivasa Ramanujan, and suggested somebody explains the “Ramanujan conjectures”. The following blog post is an attempt at an informal account. Or, as @tjf frames it, my christmas present to math twitter.

The story begins 1916, in a paper Ramanujan published in the Transactions of the Cambridge Philosophical Society, under the not so explicit title: On certain arithmetical functions. His goal started as the investigation of the sum $\sigma_s(n)$ of all $s$th powers of all divisors of an integer $n$, and approximate functional equations of the form
\[ \sigma_r(0)\sigma_s(n)+\sigma_r(1)\sigma_s(n-1)+\dots+\sigma_r(n)\sigma_s(0)
\approx \frac{\Gamma(r+1)\Gamma(s+1)}{\Gamma(r+s+2)} \frac{\zeta(r+1)\zeta(s+1)}{\zeta(r+s+2)}\sigma_{r+s+1}(n) + \frac{\zeta(1-r)+\zeta(1-s)}{r+s} n \sigma_{r+s-1}(n), \]
where $\sigma_s(0)=\dfrac12 \zeta(-s)$, and $\zeta$ is Riemann's zeta function. In what follows, $r,s$ will be positive odd integers, so that $\sigma_s(0)$ is half the value of Riemann's zeta function at a negative odd integer; it is known to be a rational number, namely $(-1)^sB_{s+1}/2(s+1)$, where $B_{s+1}$ is the $(s+1)$th Bernoulli number.

This investigation, in which Ramanujan engages without giving any motivation, quickly leads him to the introduction of infinite series,
$$S_r = \frac12 \zeta(-r) + \frac{1^rx}{1-x}+\frac{2^r x^2}{1-x^2}+\frac{3^rx^3}{1-x^3}+\dots.$$
Nowadays, the parameter $x$ would be written $q$, and $S_r=\frac12 \zeta(-r) E_{r+1}$, at least if $r$ is an odd integer, $E_r$ being the Fourier expansion of the Eisenstein series of weight $r$. The particular cases $r=1,3,5$ are given special names, namely $P,Q,R$, and Ramanujan proves that $S_s$ is a linear combination of $Q^mR^n$, for integers $m,n$ such that $4m+6n=s+1$. Nowadays, we understand this as the fact that $Q$ and $R$ generated the algebra of modular forms—for the full modular group $\mathrm{SL}(2,\mathbf Z)$.

In the same paper, Ramanujan spells out the system of algebraic differential equations satisfied by $P,Q,R$:
$$x \frac {dP}{dx} = \frac{1}{12}(P^2-Q),  x\frac{dQ}{dx}=\frac13(PQ-R), x\frac{dR}{dx}=\frac12(PR-Q^2).$$

The difference of the two sides of the initial equation has an expansion as a linear combination of $Q^mR^n$, where $4m+6n=r+s+2$. By the functional equation of Riemann's zeta function, relating $\zeta(s)$ and $\zeta(1-s)$, this expression vanishes for $x=0$, hence there is a factor $Q^3-R^2$.

Ramanujan then notes that

$x\frac{d}{dx} \log(Q^3-R^2)=P$, so that

$$P =x\frac{d}{dx} \log \left( x\big((1-x)(1-x^2)(1-x^3)\dots\big)^{24} \right)$$

and 

$$Q^3-R^2 = 1728 x  \big((1-x)(1-x^2)(1-x^3)\dots\big)^{24}= \sum \tau(n) x^n,$$
an expression now known as Ramanujan's $\Delta$-function. In fact, Ramanujan also makes the relation with elliptic functions, in particular, with Weierstrass's $\wp$-function. Then, $\Delta$ corresponds to the discriminant of the degree 3 polynomial $f$ such that $\wp'(u)^2=f(\wp(u))$. 

In any case, factoring $Q^3-R^2$ in the difference of the two terms, it is written as a linear combination of $Q^mR^n$, where $4m+6n=r+s-10$. When $r$ and $s$ are positive odd integers such that $r+s\leq 12$, there are no such pairs $(m,n)$, hence the difference vanishes, and Ramanujan obtains an equality in these cases. 

Ramanujan is interested in the quality of the initial approximation. He finds an upper bound of the form $\mathrm O(n^{\frac23(r+s+1)})$. Using Hardy–Littlewood's method, he shows that it cannot be smaller than $n^{\frac12(r+s)}$. That prompts his interest for the size of the coefficients of arithmetical functions, and $Q^3-R^2$ is the simplest one. He computes the coefficients $\tau(n)$ for $n\leq30$ and gives them in a table:

Recalling that $\tau(n)$ is $\mathrm O(n^7)$, and not $\mathrm O(n^5)$, Ramanujan states that there is reason to believe that $\tau(n)=\mathrm O(n^{\frac{11}2+\epsilon})$ but not $\mathrm O(n^{\frac{11}2})$. That this holds is Ramanujan's conjecture.

Ramanujan was led to believe this by observing that the Dirichlet series $\sum \frac{\tau(n)}{n^s}$ factors as an infinite product (“Euler product”, would we say), indexed by the prime numbers:

$$\sum_{n=1}^\infty \frac{\tau(n)}{n^s} = \prod_p \frac{1}{1-\tau(p)p^{-s}+p^{11-2s}}.$$

This would imply that $\tau$ is a multiplicative function: $\tau(mn )=\tau(m)\tau(n)$ if $m$ and $n$ are coprime, as well as the more complicated relation $\tau(p^{k+2})=\tau(p)\tau(p^{k+1})-p^{11}\tau(p^k)$ between the $\tau(p^k)$. These relations have been proved by Louis Mordell in 1917. He introduced operators (now called Hecke operators) $T_p$ (indexed by prime numbers $p$) on the algebra of modular functions and proved that Ramanujan's $\Delta$-function is an eigenfunction. (It has little merit for that, because it is alone in its weight, so that $T_p \Delta$ is a multiple of $\Delta$, necessarily $T_p\Delta=\tau(p)\Delta$.)

The bound $\lvert{\tau(p)}\rvert\leq p^{11/2}$ means that the polynomial $1-\tau(p) X+p^{11}X^2$ has two complex conjugate roots. This part of the conjecture would be proved in 1973 only, by Pierre Deligne, and required many additional ideas. One was conjectures of Weil about the number of points of algebraic varieties over finite fields, proved by Deligne in 1973, building on Grothendieck's étale cohomology. Another was the insight (due to Michio Kuga, Mikio Sato and Goro Shimura) that Ramanujan's conjecture could be reframed as an instance of the Weil conjectures, and its actual proof by Deligne in 1968, applied to the 10th symmetric product of the universal elliptic curve.