The Rankin–Selberg Method

A modular form f carries an infinite string of numbers — its Fourier coefficients a_1, a_2, a_3, \dots — and almost every deep fact about f is really a fact about those numbers. So here is a natural, greedy question. Take two forms, f with coefficients a_n and g with coefficients b_n, and multiply their coefficients together term by term. What does the Dirichlet series of the product,

\sum_{n\ge1} \frac{a_n\,\overline{b_n}}{n^{s}},

actually look like? Does it have an Euler product? A meromorphic continuation? A functional equation? And — the prize question — how big are the coefficients a_n on average?

In 1939, working independently and an ocean apart, Robert Rankin in Cambridge and Atle Selberg in Norway found the same beautiful answer, by the same trick. It is now called the Rankin–Selberg method, and it is one of the most reliable machines in the whole subject: point it at two forms and out drops a brand-new L-function, its analytic continuation, and — as a first bite at the legendary Ramanujan conjecture — a genuine bound on the size of a single coefficient.

The obstacle: multiplying coefficients is violent

Why is this hard? The L-function of a single Hecke eigenform, L(s,f) = \sum a_n n^{-s}, is lovely because the coefficients are multiplicative: they factor into an Euler product over primes, and each local factor is a tidy quadratic. But the product a_n \overline{b_n} mixes the two forms' arithmetic together. The naive series has no obvious integral representation and no obvious symmetry. You cannot just multiply the two L-functions — L(s,f)L(s,g) is the Dirichlet series of the Dirichlet convolution \sum_{d\mid n} a_d\, b_{n/d}, which is a completely different animal from the term-by-term product a_n \overline{b_n} we want.

The Rankin–Selberg insight is to stop staring at the series and instead go up into the geometry where f and g live — the upper half-plane \mathbb{H} = \{ z = x + iy : y > 0\} — and let an integral do the bookkeeping.

Both f and g are modular forms of weight k for \mathrm{SL}_2(\mathbb{Z}), with Fourier expansions f(z) = \sum_{n\ge1} a_n e^{2\pi i n z} and likewise for g. The magic ingredient is a third object: the real-analytic Eisenstein series.

The third player: the real-analytic Eisenstein series

Alongside f and g we bring in a form that carries its own complex variable s:

E(z,s) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} \operatorname{Im}(\gamma z)^{s} = \tfrac12 \sum_{\substack{(c,d)=1}} \frac{y^{s}}{|cz+d|^{2s}}.

Here \Gamma = \mathrm{SL}_2(\mathbb{Z}) and \Gamma_\infty is the subgroup of translations z \mapsto z+n. Think of E(z,s) as a "smeared out" version of the height function y^{s}, averaged over all the ways the modular group can move a point around. It converges for \Re(s) > 1, and — this is the whole point — it is already known to extend meromorphically to all s and to satisfy a functional equation relating E(z,s) to E(z,1-s), with a single simple pole at s = 1 whose residue is a constant (the reciprocal of the volume of the fundamental domain).

Everything good about the Rankin–Selberg L-function will be inherited from E(z,s). We do not have to prove continuation and functional equation for our new series from scratch — the Eisenstein series has already done that work, and we get to borrow it. That is the trick's genius.

Unfolding — the one move that makes it all work

Consider the integral of the product of all three objects over the fundamental domain \mathcal{F} = \Gamma \backslash \mathbb{H}, against the invariant measure d\mu = \dfrac{dx\,dy}{y^{2}}:

I(s) = \int_{\mathcal{F}} f(z)\,\overline{g(z)}\, y^{k}\, E(z,s)\, \frac{dx\,dy}{y^{2}}.

(The factor y^{k} makes the combination f\,\overline{g}\,y^{k} genuinely \Gamma-invariant, so the integrand descends to the quotient.) Now substitute the sum defining E(z,s) and perform the unfolding: summing over cosets \gamma \in \Gamma_\infty\backslash\Gamma and integrating over one copy of the fundamental domain is the same as not summing at all and instead integrating over the larger region \Gamma_\infty\backslash\mathbb{H} — a single vertical strip \{0 \le x < 1,\ y > 0\}. The infinite sum of translated copies of \mathcal{F} tiles that strip exactly once. The cosets fold up into a single clean domain:

I(s) = \int_{0}^{\infty}\!\!\int_{0}^{1} f(z)\,\overline{g(z)}\, y^{s+k}\, \frac{dx\,dy}{y^{2}}.

This is the miracle. A messy integral over the awkward fundamental domain has become a plain integral over a strip — and on a strip we can use Fourier series.

Picturing the unfold

The whole method rests on a picture. The fundamental domain \mathcal{F} is the classic curved region at the top; the group \Gamma stamps copies of it all across the upper half-plane. Summing E(z,s) over cosets, then folding into the quotient, is undone by unfolding: those copies reassemble into one straight vertical strip, each point covered exactly once. Step through it.

Once we are on the strip, the inner integral \int_0^1 f(z)\overline{g(z)}\,dx is exactly the operation that extracts the constant Fourier coefficient of the product — and that is where the coefficient sum \sum a_n \overline{b_n} is hiding.

Worked example: the double sum collapses to a diagonal

Let us actually turn the crank on the inner integral. Write out the Fourier expansions and multiply:

f(z)\,\overline{g(z)} = \sum_{m\ge1}\sum_{n\ge1} a_m\,\overline{b_n}\; e^{2\pi i m z}\, \overline{e^{2\pi i n z}} = \sum_{m,n} a_m\,\overline{b_n}\; e^{2\pi i (m-n)x}\, e^{-2\pi (m+n) y}.

Now integrate over x from 0 to 1. The oscillating factor e^{2\pi i (m-n)x} integrates to 0 unless m = n, in which case it integrates to 1. This is orthogonality of characters, and it is doing all the work: it forces the double sum onto its diagonal, killing every cross term and leaving only m = n:

\int_0^1 f(z)\,\overline{g(z)}\,dx = \sum_{n\ge1} a_n\,\overline{b_n}\; e^{-4\pi n y}.

Feed this back into the strip integral and do the remaining y-integral term by term. Each term is a Gamma integral — \int_0^\infty e^{-4\pi n y}\, y^{s+k-2}\,dy = \dfrac{\Gamma(s+k-1)}{(4\pi n)^{s+k-1}} — so:

I(s) = \frac{\Gamma(s+k-1)}{(4\pi)^{s+k-1}} \sum_{n\ge1} \frac{a_n\,\overline{b_n}}{n^{s+k-1}}.

There it is. The Rankin–Selberg integral equals a Gamma factor times exactly the coefficient-convolution Dirichlet series we wanted, shifted by k-1. (The y-integral is a Gamma function in disguise.) The awkward object we could not analyse directly is now expressed as a clean integral we understand completely.

The Rankin–Selberg L-function, and its free inheritance

Package the series into its own name. The Rankin–Selberg convolution L-function is

L(s, f\times g) = \zeta(2s) \sum_{n\ge1} \frac{a_n\,\overline{b_n}}{n^{s+k-1}},

where the \zeta(2s) factor is a normalisation that appears naturally alongside the coprimality condition (c,d)=1 in the Eisenstein series. Now cash in the inheritance.

We proved none of continuation or functional equation for the convolution series by hand — every bit of it is a gift from the Eisenstein series, ferried across by the unfolding identity. That is the Rankin–Selberg method in one sentence: attach your unknown series to an Eisenstein series you already understand, and inherit its analysis.

The headline payoff: the average size of the coefficients

The pole is not a nuisance — it is the punchline. When g = f, the numerator is a_n \overline{a_n} = |a_n|^{2}, and our series is the Dirichlet series of the squared magnitudes of the coefficients. A simple pole of a Dirichlet series \sum c_n n^{-s} at s = k with residue c is precisely the analytic signature of an average — by a Tauberian theorem it means \sum_{n\le x} c_n \sim \frac{c}{k}\,x^{k}. Applied here:

This n^{(k-1)/2} is exactly the size the Ramanujan conjecture predicts for every coefficient. Rankin–Selberg cannot prove it for every n — but by a clean argument it turns the average into a bound on each individual coefficient. Since a single term cannot exceed the whole sum, one gets the Rankin–Selberg bound

a_n \ll n^{\frac{k-1}{2} + \frac{1}{5} + \varepsilon}.

The extra \tfrac15 in the exponent is the "error" left by the method — the true Ramanujan answer has no extra at all. For decades this was the best unconditional bound known, the first serious approach to Ramanujan, until Pierre Deligne proved the full conjecture in 1974 as a consequence of his proof of the Weil conjectures. Rankin–Selberg got there first, by elementary-looking analysis, and it still gives the sharpest known average.

The single most important thing to keep straight: the Rankin–Selberg integral computes \sum |a_n|^{2} n^{-s} — the second moment of the coefficients — not \sum a_n n^{-s} and not individual a_n. It is fundamentally an averaging tool. The bound on a single a_n only comes out afterwards, and only because one term is dominated by the whole positive sum — which is exactly why it is lossy, leaving the stray \tfrac15 that Ramanujan/Deligne removes. If you want a genuine statement about one coefficient with no averaging, this is not your tool.

Second warning: the whole edifice needs the Eisenstein series' analytic properties. The continuation and functional equation of L(s, f\times g) are not proved by the unfolding — unfolding only produces the identity tying the two together. Continuation and the functional equation are imported from E(z,s). If the Eisenstein series did not already have a meromorphic continuation and a pole at s=1, there would be no mean-value theorem and no bound. The Eisenstein series is not scaffolding you throw away — it is the engine.

Robert Rankin published his version in 1939 in Cambridge; Atle Selberg published his the same year in Norway, on the brink of the war, with no knowledge of Rankin's work. Both were chasing the size of Ramanujan's \tau(n) function — the coefficients of the weight-12 form \Delta — and both hit on the idea of pairing the form against an Eisenstein series and integrating. Simultaneous independent discovery is a recurring theme in mathematics (Newton and Leibniz, Bolyai and Lobachevsky); it usually means the idea's time had genuinely come — the tools (Hecke's theory of modular forms, the real-analytic Eisenstein series) had just matured enough that the move became almost inevitable to anyone staring hard at the right integral.

The \tfrac15 is not arbitrary — it falls out of optimising the trade-off in the argument that passes from the average \sum_{n\le x}|a_n|^2 \sim c\,x^k to a bound on a single a_n. Roughly, you bound one coefficient by the mean over a short window and balance two competing error terms; the balance point lands the exponent at \tfrac{k-1}{2} + \tfrac15. Different, cleverer amplification arguments shave the \tfrac15 down (Rankin himself improved it later), but no purely analytic method ever reached the sharp exponent \tfrac{k-1}{2} — that needed Deligne's algebraic geometry. It is a lovely example of a "soft" method getting agonisingly close to a "hard" truth.

Why it still matters: the modern life of the method

Rankin–Selberg long ago outgrew its original job. The same unfolding template, generalised to automorphic forms on higher-rank groups, is one of the backbone constructions of the whole Langlands program. A few of its modern descendants:

Symmetric-square L-functions. Taking f \times f and stripping off the "obvious" factor \zeta(s) isolates the symmetric-square L-function L(s, \operatorname{Sym}^2 f), whose analytic properties (again inherited through the method) control finer statistics of the coefficients.

Subconvexity. Bounding L(s,f\times g) on the critical line better than the "trivial" convexity bound is a central modern problem; Rankin–Selberg integrals are the arena where much of that work happens, feeding into equidistribution results about geodesics and lattice points.

Sato–Tate. The proof that the normalised coefficients a_p / p^{(k-1)/2} follow the Sato–Tate distribution runs precisely on the analytic continuation and non-vanishing of an infinite family of symmetric-power L-functions — objects built and understood through the Rankin–Selberg machinery. The method that first estimated the size of the coefficients now helps prove how they are distributed.

The coefficients also underpin the L-functions of individual Hecke eigenforms, whose convolutions are exactly what the method takes as input. Eighty-five years on, "take two forms and integrate them against an Eisenstein series" remains a first thing to try.