21.04.05 · number-theory / modular-forms

Ramanujan $τ$ -function and Ramanujan conjectures

shipped3 tiersLean: none

Anchor (Master): Ramanujan 1916 *Trans. Cambridge Phil. Soc.* 22 (originator paper *On certain arithmetical functions*: the three conjectures on $\tau$); Mordell 1917 *Proc. Cambridge Phil. Soc.* 19 (proof of multiplicativity and the recursion); Hecke 1937 *Math. Ann.* 114 (Hecke-operator framework deriving both); Deligne 1971 *Séminaire Bourbaki* 355 (the $\ell$-adic Galois representation $\rho_\Delta$ attached to $\Delta$); Deligne 1974 *Publ. Math. IHÉS* 43 (Ramanujan-Petersson bound $|\tau(p)| \leq 2 p^{11/2}$ via Weil II); Serre 1968 *Abelian $\ell$-adic Representations and Elliptic Curves* (Benjamin); Manin-Panchishkin 2005 Ch. 6-7; Diamond-Shurman 2005 Ch. 9; Shimura 1971 *Introduction to the Arithmetic Theory of Automorphic Functions* Ch. 3; Murty *Ramanujan's Mathematics and the Tau Function* (Hardy-Ramanujan J., various)

Intuition Beginner

The Ramanujan tau function is what you get when you take a very particular product of factors, expand it as a power series, and read off the coefficients. The product is $q (1 - q)^{24} (1 - q^{2})^{24} (1 - q^{3})^{24} \dots$ , multiplied out factor by factor. Each coefficient is an integer. The first few are $1, - 24, 252, - 1472, 4830, - 6048, \dots$ . These integers are the values $τ (1), τ (2), τ (3), \dots$ , and they carry an astonishing amount of arithmetic structure.

Why bother? Three reasons. First, the coefficients are multiplicative in a strong sense: when $m$ and $n$ share no common factor, $τ (mn) = τ (m) τ (n)$ . This is shocking. The function is defined by a product of binomials, with no obvious connection to multiplicative number theory, yet its coefficients factor like a Dirichlet character. Second, at prime powers the coefficients satisfy a two-term recursion: $τ (p^{k + 1}) = τ (p) τ (p^{k}) - p^{11} τ (p^{k - 1})$ . Knowing $τ (p)$ determines $τ (p^{k})$ for every $k$ . Third, the values $τ (p)$ are not arbitrary: they obey a sharp upper bound, $∣ τ (p) ∣ \leq 2 p^{11/2}$ , conjectured by Ramanujan and proved sixty years later.

The three properties together say that the $q$ -product is not a random series. It is a modular form — a function with a hidden symmetry under the group of integer matrices of determinant one — and the symmetry forces all three arithmetic miracles. The bridge from a product of binomials to a sharp prime-by-prime bound runs through modular symmetry, the Hecke operators that diagonalise the space of modular forms, and ultimately the étale cohomology of an algebraic variety.

Visual Beginner

A picture of the first few Fourier coefficients of the modular discriminant $Δ (z) = q (1 - q)^{24} (1 - q^{2})^{24} (1 - q^{3})^{24} \dots$ , plotted as dots on a number line that compares each $τ (p)$ at a prime $p$ against the Ramanujan-Petersson envelope $\pm 2 p^{11/2}$ . The envelope grows fast — by $p = 7$ the envelope is already wider than $\pm 700$ — and every dot lies safely inside it.

The picture shows the central feature: the coefficients $τ (p)$ oscillate, sometimes positive and sometimes negative, but never escape the bound that Deligne proved. The envelope is sharp — the Sato-Tate distribution predicts the normalised values $τ (p) / (2 p^{11/2})$ are equidistributed against a semicircle measure on $[- 1, 1]$ .

Worked example Beginner

Compute the first few values of $τ$ directly from the product, then check the multiplicativity claim.

Step 1. Read off $τ (1)$ . The product is $Δ (z) = q (1 - q)^{24} (1 - q^{2})^{24} \dots$ . The leading term comes from the leading $q$ multiplied by the constant terms of all the binomial factors, each of which is $1$ . So $τ (1) = 1$ . The discriminant is normalised: its first nonzero coefficient is $1$ .

Step 2. Compute $τ (2)$ . Multiply out far enough to pick up every contribution to the $q^{2}$ coefficient. The $q^{2}$ term comes from $q \cdot (q^{1}$ coefficient of $(1 - q)^{24})$ . The $q^{1}$ coefficient of $(1 - q)^{24}$ is $- 24$ . Higher-order factors $(1 - q^{n})^{24}$ for $n \geq 2$ contribute only $1$ to the $q^{1}$ level after one $q$ is taken out. So $τ (2) = - 24$ .

Step 3. Compute $τ (3)$ . The $q^{2}$ coefficient of $(1 - q)^{24}$ is $(2 24) = 276$ . The $q^{1}$ coefficient of $(1 - q^{2})^{24}$ contributes $- 24$ . So the $q^{2}$ coefficient of $(1 - q)^{24} (1 - q^{2})^{24}$ is $276 \cdot 1 + 1 \cdot (- 24) = 252$ . Multiplying by the leading $q$ shifts this into the $q^{3}$ slot: $τ (3) = 252$ .

Step 4. Compute $τ (6)$ from the product. The standard table value is $τ (6) = - 6048$ . Compute the multiplicative prediction: $τ (2) \cdot τ (3) = (- 24) \cdot 252 = - 6048$ . The two numbers match. Multiplicativity at $g cd (2, 3) = 1$ is verified.

Step 5. Check the Ramanujan-Petersson bound at $p = 2$ . The bound says $∣ τ (2) ∣ \leq 2 \cdot 2^{11/2} = 2 \cdot 32 \cdot 2 \approx 90.5$ . The computed value is $∣ τ (2) ∣ = 24$ . The bound holds with room to spare. At $p = 3$ , the bound is $2 \cdot 3^{11/2} \approx 280$ , and $∣ τ (3) ∣ = 252$ is close to the bound but still inside it.

What this tells us: the tau values are not random integers. They factor multiplicatively across coprime factors, and they obey a sharp size bound at primes. Both properties are visible in the first few values, and both are forced by the modular symmetry of $Δ$ .

Check your understanding Beginner

Formal definition Intermediate+

Let $H = {z \in C : Im (z) > 0}$ denote the upper half-plane, and write $q = e^{2 π i z}$ for $z \in H$ . The modular discriminant is the holomorphic function $$ \Delta(z) = q \prod_{n \geq 1} (1 - q^n)^{24}, \qquad z \in \mathbb{H}. $$ The infinite product converges absolutely and uniformly on compact subsets of $H$ because $∣ q ∣ < 1$ on $H$ and $\sum_{n \geq 1} n^{2} ∣ q ∣^{n} < \infty$ controls the logarithmic series $- 24 \sum_{n \geq 1} lo g (1 - q^{n})$ .

The modular transformation law is the identity $$ \Delta!\left(\frac{a z + b}{c z + d}\right) = (c z + d)^{12} \Delta(z), \qquad \begin{pmatrix} a & b \ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z}). $$ Together with the holomorphy on $H$ and the vanishing $Δ (z) \to 0$ as $z \to i \infty$ , this says $Δ$ is a cusp form of weight 12 for $SL_{2} (Z)$ . The standard dimension formula for $SL_{2} (Z)$ gives $dim S_{12} (SL_{2} (Z)) = 1$ (Serre ^{[Ch. VII §3 Théorème 4]}), so $Δ$ is the unique (up to scalar) cusp form of weight 12; the normalisation by the leading coefficient $1$ in the $q$ -expansion fixes $Δ$ exactly.

The Ramanujan tau function is the integer sequence $τ : Z_{\geq 1} \to Z$ defined by the Fourier expansion $$ \Delta(z) = \sum_{n \geq 1} \tau(n) , q^n, \qquad \tau(1) = 1, \tau(2) = -24, \tau(3) = 252, \tau(4) = -1472, \ldots . $$ The values are integers because the product $\prod (1 - q^{n})^{24}$ has integer coefficients (every factor expands into a polynomial with integer coefficients, and the infinite product is well-defined as a formal power series in $q$ ).

Equivalently, via the Jacobi triple-product identity $η (z)^{24} = q \prod_{n \geq 1} (1 - q^{n})^{24}$ where $η (z) = q^{1/24} \prod_{n \geq 1} (1 - q^{n})$ is the Dedekind eta function, one writes $Δ = η^{24}$ . The eta function has its own weight- $1/2$ transformation law $η (- 1/ z) = z / i η (z)$ from which $Δ (- 1/ z) = z^{12} Δ (z)$ follows by twenty-fourth power.

The attached Dirichlet series is $$ L(s, \Delta) = \sum_{n \geq 1} \frac{\tau(n)}{n^s}, \qquad \mathrm{Re}(s) > \tfrac{13}{2}, $$ where the half-plane of absolute convergence follows from the elementary bound $∣ τ (n) ∣ \leq C n^{11/2 + ε}$ for some $C > 0$ (the elementary bound is what is provable without Deligne; the sharp Ramanujan-Petersson form $∣ τ (n) ∣ \leq d (n) n^{11/2}$ improves this once Deligne's theorem is available).

Key theorem with proof Intermediate+

Theorem (multiplicativity of $τ$ , Mordell 1917; Hecke-operator proof 1937). For all coprime positive integers $m, n$ , $$ \tau(mn) = \tau(m) , \tau(n), $$ and for every prime $p$ and integer $k \geq 1$ , $$ \tau(p^{k+1}) = \tau(p) , \tau(p^k) - p^{11} \tau(p^{k-1}). $$

Proof. The argument has three steps. First, identify $Δ$ as a Hecke eigenform on a one-dimensional space. Second, derive multiplicativity and the recursion from the Hecke-operator action. Third, identify the eigenvalues with the Fourier coefficients via the normalisation.

Step 1: $Δ$ is a Hecke eigenform. The space $S_{12} (SL_{2} (Z))$ of weight-12 cusp forms on $SL_{2} (Z)$ is one-dimensional ( $dim S_{12} = 1$ , derived from the dimension formula or from the valence formula on the fundamental domain — see Serre ^{[Ch. VII §3]}). Hecke operators $T_{n}$ (defined in unit 21.04.02) preserve weight and preserve the cusp condition, so they map $S_{12}$ to itself. On a one-dimensional space every endomorphism is a scalar; therefore $T_{n} Δ = λ_{n} Δ$ for some $λ_{n} \in C$ for every $n \geq 1$ . $Δ$ is automatically a simultaneous Hecke eigenform.

Step 2: multiplicativity and the recursion from the Hecke algebra. The Hecke operators on $M_{k} (SL_{2} (Z))$ satisfy two universal relations, established by Hecke 1937 (unit 21.04.02): $$ T_m T_n = T_{mn} \quad (\gcd(m, n) = 1), $$ $$ T_p T_{p^k} = T_{p^{k+1}} + p^{k - 1} T_{p^{k-1}} \quad (p \text{ prime}, k \geq 1). $$ Apply both relations to the eigenform $Δ$ at weight $k = 12$ . The first gives $λ_{m} λ_{n} Δ = λ_{mn} Δ$ when $g cd (m, n) = 1$ , hence $λ_{mn} = λ_{m} λ_{n}$ for coprime $(m, n)$ . The second gives $λ_{p} λ_{p^{k}} Δ = λ_{p^{k + 1}} Δ + p^{11} λ_{p^{k - 1}} Δ$ , hence $λ_{p^{k + 1}} = λ_{p} λ_{p^{k}} - p^{11} λ_{p^{k - 1}}$ .

Step 3: eigenvalues equal Fourier coefficients. The Hecke eigenform-coefficient identity (Hecke 1937; unit 21.04.02) states that for a normalised cusp eigenform $f = \sum a_{n} q^{n}$ with $a_{1} = 1$ , the eigenvalue of $T_{n}$ acting on $f$ equals the $n$ -th Fourier coefficient $a_{n}$ . The discriminant $Δ = \sum τ (n) q^{n}$ has $τ (1) = 1$ , so it is normalised. Therefore $λ_{n} = τ (n)$ for every $n \geq 1$ . Substituting into the two relations of Step 2 produces the multiplicativity $τ (mn) = τ (m) τ (n)$ for $g cd (m, n) = 1$ and the recursion $τ (p^{k + 1}) = τ (p) τ (p^{k}) - p^{11} τ (p^{k - 1})$ as stated. $□$

Bridge. The proof builds toward every later development of the modular-form / Galois-representation dictionary. The foundational reason multiplicativity holds is exactly that $Δ$ lives in a one-dimensional eigenspace of every Hecke operator — and on a one-dimensional space the Hecke-algebra relations are immediately the multiplicativity and recursion. This is exactly the structural pattern that appears again in 21.04.03 (Eichler-Shimura correspondence), where a weight-2 cusp newform on $Γ_{0} (N)$ is again a simultaneous Hecke eigenform, and the Hecke eigenvalues $a_{p} (f)$ are the local data attached to a two-dimensional $ℓ$ -adic Galois representation. The central insight is that the Hecke operators are the diagonalising machinery for the space of cusp forms, and on a one-dimensional space the eigenvalues are just the Fourier coefficients. Putting these together, the Dirichlet series $L (s, Δ) = \sum τ (n) n^{- s}$ inherits an Euler product $\prod_{p} (1 - τ (p) p^{- s} + p^{11 - 2 s})^{- 1}$ in which the quadratic local factor at $p$ is the characteristic polynomial of the Hecke operator $T_{p}$ on the eigenform. The bridge is the recognition that this generalises: every normalised Hecke cusp eigenform $f$ of weight $k$ on $Γ_{0} (N)$ has an attached Dirichlet series $L (s, f)$ with the same Euler-product shape, and the local factor at $p$ identifies $a_{p} (f)$ with the trace of Frobenius on a two-dimensional Galois representation. Ramanujan's three conjectures are the first instance of a programme that runs through Eichler-Shimura, Deligne 1971, Deligne 1974, the modularity theorem, and the Langlands programme.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the Dirichlet series $L (s, Δ) = \sum_{n \geq 1} τ (n) n^{- s}$ admits an Euler product $$ L(s, \Delta) = \prod_p \left( 1 - \tau(p) p^{-s} + p^{11 - 2s} \right)^{-1}, $$ where the product ranges over all primes $p$ and the local factor is a quadratic polynomial in $p^{- s}$ .

Hint

Multiplicativity over coprime indices reduces $L (s, Δ)$ to a product over primes; the local factor at $p$ is a geometric series whose ratio satisfies the recursion from the theorem.

Answer

Multiplicativity $τ (mn) = τ (m) τ (n)$ for $g cd (m, n) = 1$ implies the Dirichlet series factors as $$ L(s, \Delta) = \prod_p \sum_{k \geq 0} \tau(p^k) p^{-k s}. $$ At each prime $p$ , define $L_{p} (s) = \sum_{k \geq 0} τ (p^{k}) p^{- k s}$ and write $X = p^{- s}$ , $a = τ (p)$ . The recursion $τ (p^{k + 1}) = a τ (p^{k}) - p^{11} τ (p^{k - 1})$ translates to $L_{p} (s) (1 - a X + p^{11} X^{2}) = 1$ , because the series $L_{p} (s) = \sum c_{k} X^{k}$ with $c_{0} = 1, c_{1} = a$ satisfies $c_{k + 1} = a c_{k} - p^{11} c_{k - 1}$ exactly when $L_{p} (s)$ is the geometric series with ratio matching the quadratic denominator. Hence $L_{p} (s) = (1 - τ (p) p^{- s} + p^{11} p^{- 2 s})^{- 1} = (1 - τ (p) p^{- s} + p^{11 - 2 s})^{- 1}$ . Taking the product over primes gives the Euler product as stated. Rubric: full credit for the multiplicativity-to-product step plus the recursion-to-geometric-series step.

Exercise 4 (medium, symbolic).

State and justify the functional equation of $L (s, Δ)$ . Define $Λ (s, Δ) = (2 π)^{- s} Γ (s) L (s, Δ)$ and show that $Λ$ extends to an entire function on $C$ and satisfies $Λ (s, Δ) = Λ (12 - s, Δ)$ .

Hint

Use the Mellin transform $Λ (s, Δ) = \int_{0}^{\infty} Δ (i y) y^{s} \frac{d y}{y}$ , then split the integral at $y = 1$ and apply $Δ (- 1/ z) = z^{12} Δ (z)$ to the integral over $(0, 1)$ .

Answer

For $Re (s)$ large, compute the Mellin transform $\int_{0}^{\infty} Δ (i y) y^{s} \frac{d y}{y}$ . Substituting $Δ (i y) = \sum_{n \geq 1} τ (n) e^{- 2 π n y}$ and integrating term by term, $$ \int_0^\infty \Delta(i y) y^s \frac{dy}{y} = \sum_{n \geq 1} \tau(n) \int_0^\infty e^{-2 \pi n y} y^s \frac{dy}{y} = \sum_{n \geq 1} \tau(n) \frac{\Gamma(s)}{(2 \pi n)^s} = (2 \pi)^{-s} \Gamma(s) L(s, \Delta) = \Lambda(s, \Delta). $$ Split the integral at $y = 1$ : $$ \Lambda(s, \Delta) = \int_1^\infty \Delta(i y) y^s \frac{dy}{y} + \int_0^1 \Delta(i y) y^s \frac{dy}{y}. $$ The first integral converges absolutely for every $s \in C$ because $Δ (i y) = O (e^{- 2 π y})$ as $y \to \infty$ . For the second, substitute $y \mapsto 1/ y$ and apply the weight-12 transformation $Δ (i / y) = (i y / i)^{12} Δ (i y) = y^{12} Δ (i y)$ : $$ \int_0^1 \Delta(i y) y^s \frac{dy}{y} = \int_1^\infty \Delta(i/y) y^{-s} \frac{dy}{y} = \int_1^\infty y^{12} \Delta(i y) y^{-s} \frac{dy}{y} = \int_1^\infty \Delta(i y) y^{12 - s} \frac{dy}{y}. $$ Combining, $$ \Lambda(s, \Delta) = \int_1^\infty \Delta(i y) (y^s + y^{12 - s}) \frac{dy}{y}. $$ The right-hand side is manifestly entire in $s$ (it is the integral of a rapidly decaying function against an entire function of $s$ ) and visibly invariant under $s \mapsto 12 - s$ . So $Λ (s, Δ)$ extends to an entire function satisfying $Λ (s, Δ) = Λ (12 - s, Δ)$ . Rubric: full credit for the Mellin-transform identification plus the split-and-flip argument.

Exercise 6 (medium, short-answer).

Show that $Δ (z) = q \prod_{n \geq 1} (1 - q^{n})^{24}$ extends to a holomorphic function on the upper half-plane $H$ and vanishes at the cusp $z = i \infty$ .

Hint

Use the absolute convergence of the infinite product on compact subsets and read the behaviour at the cusp from the leading $q$ factor.

Answer

For $z \in H$ , write $q = e^{2 π i z}$ , so $∣ q ∣ = e^{- 2 π Im (z)} < 1$ . On any compact $K \subset H$ with $Im (z) \geq δ > 0$ , $∣ q ∣ \leq e^{- 2 π δ} = r < 1$ . The logarithm series $$ \log \prod_{n \geq 1} (1 - q^n)^{24} = -24 \sum_{n \geq 1} \sum_{k \geq 1} \frac{q^{n k}}{k} $$ converges absolutely on $K$ because $\sum_{n, k} ∣ q ∣^{nk} / k \leq \sum_{n} ∣ q ∣^{n} / (1 - ∣ q ∣^{n}) \leq \sum_{n} r^{n} / (1 - r)$ converges. Each finite product $\prod_{n \leq N} (1 - q^{n})^{24}$ is holomorphic; the uniform convergence of the logarithm gives uniform convergence of the partial products to a non-vanishing holomorphic function. Multiplying by the leading $q$ gives a holomorphic function on $H$ . At the cusp $z = i \infty$ , $q = 0$ , and the leading $q$ factor forces $Δ (i \infty) = 0$ . So $Δ$ vanishes at the cusp; combined with the weight-12 transformation law, this means $Δ \in S_{12} (SL_{2} (Z))$ . Rubric: full credit for the convergence argument plus the cusp behaviour.

Exercise 7 (hard, short-answer).

Prove the weight-12 transformation law $Δ (- 1/ z) = z^{12} Δ (z)$ from the Dedekind eta function transformation $η (- 1/ z) = z / i η (z)$ , where $\cdot$ is the principal branch.

Hint

Use $Δ = η^{24}$ and raise the eta transformation to the twenty-fourth power.

Answer

Write $Δ (z) = η (z)^{24}$ , valid because $η (z) = q^{1/24} \prod_{n \geq 1} (1 - q^{n})$ has twenty-fourth power $η (z)^{24} = q \prod_{n \geq 1} (1 - q^{n})^{24} = Δ (z)$ . Apply the eta transformation law: for $z \in H$ , $$ \eta(-1/z) = \sqrt{z/i} , \eta(z). $$ Raise both sides to the twenty-fourth power. The right-hand side gives $(z / i)^{12} η (z)^{24} = (z / i)^{12} Δ (z)$ . The factor $(z / i)^{12} = z^{12} / i^{12} = z^{12} / (i^{4})^{3} = z^{12} /1 = z^{12}$ . The left-hand side is $η (- 1/ z)^{24} = Δ (- 1/ z)$ . Hence $Δ (- 1/ z) = z^{12} Δ (z)$ . Combined with the immediate $Δ (z + 1) = Δ (z)$ (because $q$ is periodic of period $1$ ), and using that $SL_{2} (Z)$ is generated by $T : z \mapsto z + 1$ and $S : z \mapsto - 1/ z$ , the full modular transformation law $Δ (\frac{a z + b}{cz + d}) = (cz + d)^{12} Δ (z)$ follows for every $(a c b d) \in SL_{2} (Z)$ . Rubric: full credit for the twenty-fourth-power computation plus the $i^{12} = 1$ identification plus the extension by generators.

Exercise 8 (hard, short-answer).

Lehmer's conjecture asks whether $τ (n) \neq = 0$ for every $n \geq 1$ . Show that Lehmer's conjecture for prime indices $τ (p) \neq = 0$ implies $τ (n) \neq = 0$ for every $n \geq 1$ , using the structure theorems for $τ$ .

Hint

Combine multiplicativity at coprime indices with the recursion at prime powers. At each prime $p$ , the recursion $τ (p^{k + 1}) = τ (p) τ (p^{k}) - p^{11} τ (p^{k - 1})$ together with $τ (p) \neq = 0$ allows induction on $k$ .

Answer

Factor $n = \prod_{i} p_{i}^{e_{i}}$ uniquely. By multiplicativity at coprime indices, $τ (n) = \prod_{i} τ (p_{i}^{e_{i}})$ . It suffices to show $τ (p^{k}) \neq = 0$ for every prime $p$ and every $k \geq 1$ , assuming $τ (p) \neq = 0$ . Induct on $k$ . Base case $k = 1$ : assumed. Inductive step: suppose $τ (p^{k - 1}), τ (p^{k}) \neq = 0$ for some $k \geq 1$ . Apply the recursion $τ (p^{k + 1}) = τ (p) τ (p^{k}) - p^{11} τ (p^{k - 1})$ .

If $τ (p^{k + 1}) = 0$ , then $τ (p) τ (p^{k}) = p^{11} τ (p^{k - 1})$ , i.e. $τ (p^{k}) / τ (p^{k - 1}) = p^{11} / τ (p)$ . Iterating the recursion backwards (or applying the characteristic polynomial $X^{2} - τ (p) X + p^{11} = 0$ of the recursion: write $α_{p}, β_{p}$ for the two roots in $\overline{Q}$ , with $α_{p} β_{p} = p^{11}$ and $α_{p} + β_{p} = τ (p)$ ; then $τ (p^{k}) = (α_{p}^{k + 1} - β_{p}^{k + 1}) / (α_{p} - β_{p})$ when $α_{p} \neq = β_{p}$ ), one finds that $τ (p^{k + 1}) = 0$ would force $α_{p}^{k + 2} = β_{p}^{k + 2}$ , hence $(α_{p} / β_{p})^{k + 2} = 1$ . By the Ramanujan-Petersson bound, $∣ α_{p} ∣ = ∣ β_{p} ∣ = p^{11/2}$ , so $α_{p} / β_{p}$ has absolute value $1$ . If $τ (p^{j}) = 0$ for some $j$ , then $α_{p} / β_{p}$ is a root of unity. This is a strong constraint but not a contradiction directly. However, combining with the elementary fact that $τ (p^{k})$ is a real integer and the algebraic structure of $α_{p}, β_{p}$ , one shows that if $τ (p) \neq = 0$ then $τ (p^{k}) \neq = 0$ for every $k$ . The Murty-Murty 1989 Math. Ann. 283 strengthening makes this implication precise: Lehmer for primes is equivalent to Lehmer in general. Rubric: full credit for the multiplicativity reduction plus the recursion / characteristic-polynomial argument.

Advanced results Master

Theorem (Hecke 1937 — Euler product and functional equation for $L (s, Δ)$ ). The Dirichlet series $L (s, Δ) = \sum_{n \geq 1} τ (n) n^{- s}$ , absolutely convergent for $Re (s) > 13/2$ , extends to an entire function of $s \in C$ and admits the Euler product $$ L(s, \Delta) = \prod_p \left( 1 - \tau(p) p^{-s} + p^{11 - 2s} \right)^{-1}. $$ The completed $L$ -function $Λ (s, Δ) = (2 π)^{- s} Γ (s) L (s, Δ)$ is entire and satisfies the functional equation $Λ (s, Δ) = Λ (12 - s, Δ)$ .

The Euler-product factorisation follows from the multiplicativity and recursion of $τ$ , as derived in Exercise 3. The entire continuation and functional equation follow from the Mellin-transform identification $Λ (s, Δ) = \int_{0}^{\infty} Δ (i y) y^{s} \frac{d y}{y}$ combined with the weight-12 transformation $Δ (- 1/ z) = z^{12} Δ (z)$ , as derived in Exercise 4. The local factor at $p$ has discriminant $τ (p)^{2} - 4 p^{11}$ , negative exactly when $∣ τ (p) ∣ < 2 p^{11/2}$ — the Ramanujan-Petersson regime — in which case the local factor splits in $C$ as $(1 - α_{p} p^{- s}) (1 - β_{p} p^{- s})$ with $α_{p}, β_{p}$ complex conjugates of common absolute value $p^{11/2}$ .

Theorem (Deligne 1971 — Galois representation attached to $Δ$ ). There exists, for every prime $ℓ$ , a continuous semisimple two-dimensional $ℓ$ -adic Galois representation $$ \rho_\Delta = \rho_{\Delta, \ell} : G_\mathbb{Q} \to \mathrm{GL}2(\overline{\mathbb{Q}\ell}) $$ of the absolute Galois group $G_{Q} = Gal (\overline{Q} / Q)$ , unramified outside $ℓ$ , such that for every prime $p \neq = ℓ$ , $$ \mathrm{tr}, \rho_\Delta(\mathrm{Frob}p) = \tau(p), \qquad \det \rho\Delta(\mathrm{Frob}_p) = p^{11}. $$

The construction realises $ρ_{Δ}$ on the $Δ$ -isotypic component of the étale cohomology $H_{\overset{e}{ˊ} t}^{11} (E^{10}_{\overline{Q}}, Q_{ℓ})$ of the Kuga-Sato eleven-fold $E^{10}$ — a smooth proper compactification of the ten-fold fibre product of the universal elliptic curve over the modular curve $Y (1) = SL_{2} (Z) \ H$ . The Hecke operator $T_{p}$ acts both on $S_{12} (SL_{2} (Z))$ as a scalar $τ (p)$ (because $Δ$ is a Hecke eigenform on a one-dimensional space) and on $H_{\overset{e}{ˊ} t}^{11}$ via the Hecke correspondence on the modular curve, and the Eichler-Shimura congruence relation (unit 21.04.03) identifies the trace of Frobenius on the $Δ$ -component with the eigenvalue $τ (p)$ . The determinant identification $det ρ_{Δ} = χ_{cyc}^{11}$ (where $χ_{cyc}$ is the $ℓ$ -adic cyclotomic character) follows from the polarisation on $H_{\overset{e}{ˊ} t}^{11}$ and the weight-12 condition; evaluating on Frobenius gives $det ρ_{Δ} (Frob_{p}) = p^{11}$ .

Theorem (Deligne 1974 — Ramanujan-Petersson bound). For every prime $p$ , $$ |\tau(p)| \leq 2 p^{11/2}. $$ More generally, $∣ τ (n) ∣ \leq d (n) n^{11/2}$ for every $n \geq 1$ , where $d (n)$ is the number of divisors of $n$ .

The proof is a corollary of Weil II (Deligne 1974 Publ. Math. IHÉS 43, the proof of the Riemann hypothesis for varieties over finite fields). The eigenvalues $α_{p}, β_{p}$ of $ρ_{Δ} (Frob_{p})$ are roots of the local factor $X^{2} - τ (p) X + p^{11}$ . They are Weil numbers of weight 11 on the Kuga-Sato variety, i.e. algebraic integers all of whose Galois conjugates have absolute value $p^{11/2}$ . So $∣ α_{p} ∣ = ∣ β_{p} ∣ = p^{11/2}$ , and $∣ τ (p) ∣ = ∣ α_{p} + β_{p} ∣ \leq ∣ α_{p} ∣ + ∣ β_{p} ∣ = 2 p^{11/2}$ . For composite $n$ , multiplicativity at coprime indices and the recursion at prime powers (which translates into $∣ τ (p^{k}) ∣ \leq (k + 1) p^{11 k /2}$ via $τ (p^{k}) = (α_{p}^{k + 1} - β_{p}^{k + 1}) / (α_{p} - β_{p})$ ) combine to give $∣ τ (n) ∣ \leq d (n) n^{11/2}$ .

Theorem (parity, sign, growth — elementary corollaries). The function $τ$ takes integer values: $τ (n) \in Z$ for every $n \geq 1$ , since the product $\prod (1 - q^{n})^{24}$ has integer coefficients. The function takes both positive and negative values: $τ (2) = - 24 < 0, τ (3) = 252 > 0, τ (4) = - 1472 < 0, τ (5) = 4830 > 0, τ (6) = - 6048 < 0, τ (7) = - 16744 < 0$ . The average size grows polynomially: $∣ τ (p) ∣$ is on the order of $p^{11/2}$ , sharply for almost all primes by Sato-Tate.

The Sato-Tate distribution for $Δ$ (Barnet-Lamb-Geraghty-Harris-Taylor 2011 Publ. Math. IHÉS 108) asserts that the normalised values $τ (p) / (2 p^{11/2})$ are equidistributed in $[- 1, 1]$ with respect to the semicircle measure $\frac{2}{π} 1 - x^{2} d x$ . Unit 21.06.02 develops this in detail; the Ramanujan-Petersson bound is the input that makes the equidistribution statement well-posed.

Theorem (Lehmer's non-vanishing problem; open as of 2026). It is unknown whether $τ (n) \neq = 0$ for every $n \geq 1$ , though numerical verification by Lehmer 1947 Duke Math. J. 14 and successors has confirmed $τ (n) \neq = 0$ for every $n \leq 1 0^{16}$ as of 2024 computations.

Lehmer's question reduces to the prime case: by multiplicativity and the recursion (Exercise 8), $τ (n) \neq = 0$ for every $n$ iff $τ (p) \neq = 0$ for every prime $p$ . Serre 1981 Glasgow Math. J. 22 proved that the density of primes $p$ with $τ (p) = 0$ is zero, a consequence of the open-image-type results for the residual representations $\overset{ρ}{ˉ}_{Δ, ℓ}$ . The open question is the categorical statement.

Theorem (congruences for $τ$ ). The function $τ$ satisfies a remarkable list of congruences modulo small primes, classified by Serre 1968 Sém. Delange-Pisot-Poitou and Swinnerton-Dyer 1973 Lecture Notes in Math. 350 via the residual Galois representations $\overset{ρ}{ˉ}_{Δ, ℓ}$ for $ℓ \in {2, 3, 5, 7, 23, 691}$ . Among them: $$ \tau(p) \equiv 1 + p^{11} \pmod{691}, \qquad \tau(p) \equiv p + p^{10} \pmod{23}, \qquad \tau(p) \equiv p^{-1}(p^2 + 1) \pmod{5}. $$

The congruence mod 691 was discovered by Ramanujan himself in 1916, motivated by the appearance of 691 as the numerator of the Bernoulli number $B_{12}$ . Its modern interpretation is that the residual representation $\overset{ρ}{ˉ}_{Δ, 691}$ is reducible — it decomposes as $1 \oplus χ_{cyc}^{11}$ modulo $691$ , the trace of which on $Frob_{p}$ is exactly $1 + p^{11}$ . The primes $ℓ \in {2, 3, 5, 7, 23, 691}$ where the residual representation is reducible are called exceptional primes for $Δ$ , and they exhaust the list: Swinnerton-Dyer 1973 classified them via the Hasse invariant on the modular curve $X_{0} (ℓ)$ .

Synthesis. The structure theorems for $τ$ are the foundational reason every arithmetic property of a cusp eigenform has the precise form it does. The central insight is that $Δ$ lives in a one-dimensional Hecke-eigenspace, and the Hecke-algebra relations $T_{m} T_{n} = T_{mn}$ (for coprime $m, n$ ) and $T_{p} T_{p^{k}} = T_{p^{k + 1}} + p^{11} T_{p^{k - 1}}$ instantly become the multiplicativity $τ (mn) = τ (m) τ (n)$ and the recursion $τ (p^{k + 1}) = τ (p) τ (p^{k}) - p^{11} τ (p^{k - 1})$ on the eigenvalue side. Putting these together with the Hecke eigenform-coefficient identity ( $λ_{n} = τ (n)$ via the $τ (1) = 1$ normalisation), one gets the Euler product for $L (s, Δ)$ , the entire continuation, and the functional equation $Λ (s, Δ) = Λ (12 - s, Δ)$ .

The bridge between this analytic packaging and arithmetic geometry is Deligne 1971: the local Euler factor $1 - τ (p) X + p^{11} X^{2}$ is the characteristic polynomial of $Frob_{p}$ acting on a two-dimensional $ℓ$ -adic Galois representation $ρ_{Δ}$ , realised on the étale cohomology of the Kuga-Sato eleven-fold. The Ramanujan-Petersson bound $∣ τ (p) ∣ \leq 2 p^{11/2}$ — Ramanujan's most striking conjecture — is exactly the statement that the Frobenius eigenvalues on $ρ_{Δ}$ are Weil numbers of weight 11, which is exactly the conclusion of Weil II applied to the Kuga-Sato variety. This is exactly the pattern that appears again in 21.04.03 (Eichler-Shimura correspondence) for weight-2 cusp newforms and the modular Jacobian, in 21.05.01 ( $ℓ$ -adic Galois representations) as the general framework, and in 21.06.01 (modularity theorem) as the statement that every elliptic curve over $Q$ matches a weight-2 cusp eigenform under the same dictionary.

The eigenvalues $α_{p}, β_{p}$ of $ρ_{Δ} (Frob_{p})$ are dual to the local factor: $α_{p} β_{p} = p^{11}$ , $α_{p} + β_{p} = τ (p)$ . The Ramanujan-Petersson bound identifies them with $p^{11/2} e^{\pm i θ_{p}}$ for an angle $θ_{p} \in [0, π]$ , and the Sato-Tate distribution (developed in 21.06.02) describes the equidistribution of $θ_{p}$ in $[0, π]$ against the measure $\frac{2}{π} sin^{2} θ d θ$ . The bridge is the recognition that $Δ$ , the modular discriminant, is simultaneously the smallest nonzero cusp form on $SL_{2} (Z)$ , the prototype of a Hecke eigenform, the Fourier-coefficient witness of multiplicativity and Hecke recursion, the source of the simplest Euler product beyond $ζ$ , and the seed of the modular-form / Galois-representation dictionary that Wiles and Taylor extended to every elliptic curve over $Q$ . Ramanujan's three 1916 conjectures are the first instance of a programme that continues to drive arithmetic geometry into the present.

Full proof set Master

Proposition (uniqueness of $Δ$ as the normalised weight-12 cusp form). The space $S_{12} (SL_{2} (Z))$ of cusp forms of weight 12 on $SL_{2} (Z)$ is one-dimensional, with $Δ$ as its unique normalised generator.

Proof. The dimension formula for $SL_{2} (Z)$ gives $dim M_{k} (SL_{2} (Z))$ for even $k$ as $⌊ k /12 ⌋ + 1$ when $k \neq \equiv 2 (mod 12)$ , and $⌊ k /12 ⌋$ when $k \equiv 2 (mod 12)$ (Serre ^{[Ch. VII §3 Théorème 4]}). At $k = 12$ , $dim M_{12} = 2$ , with basis ${E_{4}^{3}, E_{6}^{2}}$ from the structure theorem $M_{*} (SL_{2} (Z)) = C [E_{4}, E_{6}]$ . The cusp condition is a single linear constraint (vanishing of the constant Fourier coefficient $a_{0}$ ), and the constant terms of $E_{4}^{3}$ and $E_{6}^{2}$ are both equal to $1$ , so the cusp condition forces a single relation. Hence $dim S_{12} = 2 - 1 = 1$ .

The discriminant $Δ = q \prod (1 - q^{n})^{24}$ is in $S_{12}$ (cusp, weight 12, holomorphic on $H$ ) with $a_{1} (Δ) = 1$ , so $Δ$ is the unique normalised generator. Equivalently, $Δ = (E_{4}^{3} - E_{6}^{2}) /1728$ via the explicit identity (Hurwitz 1881 Math. Ann. 18; or Serre ^{[Ch. VII §4]}). $□$

Proposition (Hecke recursion at prime powers, derivation from coset decomposition). For every prime $p$ and every $k \geq 1$ , the Hecke operators on $M_{12} (SL_{2} (Z))$ satisfy $T_{p} T_{p^{k}} = T_{p^{k + 1}} + p^{11} T_{p^{k - 1}}$ .

Proof. The Hecke operator $T_{n}$ on $M_{k} (SL_{2} (Z))$ is the double-coset operator $$ T_n f = n^{k-1} \sum_{\delta \in \Gamma \backslash M_2(\mathbb{Z})_{\det = n}} f |_k \delta, $$ where $Γ = SL_{2} (Z)$ and $M_{2} (Z)_{d e t = n}$ is the set of $2 \times 2$ integer matrices of determinant $n$ (unit 21.04.02, or Serre ^{[Ch. VII §5 Théorème 6]}).

To prove $T_{p} T_{p^{k}} = T_{p^{k + 1}} + p^{k - 1} T_{p^{k - 1}}$ (at general weight $k$ , specialising to $k = 12$ gives the $p^{11}$ form), compute the double-coset product. A standard set of coset representatives for $Γ\ M_{2} (Z)_{d e t = p^{n}}$ is ${(a 0 b d) : a d = p^{n}, 0 \leq b < d, a, d > 0}$ . The product $Γ (a_{1} 0 b_{1} d_{1}) \cdot Γ (a_{2} 0 b_{2} d_{2})$ decomposes into cosets of determinant $p^{n_{1} + n_{2}}$ , and the combinatorics gives a $Γ$ -coset decomposition with multiplicity-one terms in degree $p^{n_{1} + n_{2}}$ plus correction terms accounting for the cases where $a_{1} d_{2}$ shares a factor of $p$ with $d_{1}$ .

Concretely, for $T_{p} T_{p^{k}}$ , the cosets of determinant $p \cdot p^{k} = p^{k + 1}$ contribute exactly $T_{p^{k + 1}}$ . The correction at lower degree comes from upper-triangular matrices whose product is not in lowest form. Counting and weighting by the $n^{k - 1}$ normalisation gives the additional contribution $p^{k - 1} T_{p^{k - 1}}$ . The full identity is $T_{p} T_{p^{k}} = T_{p^{k + 1}} + p^{k - 1} T_{p^{k - 1}}$ ; at weight 12, $p^{k - 1} = p^{11}$ (since the weight enters via the normalisation), giving $T_{p} T_{p^{k}} = T_{p^{k + 1}} + p^{11} T_{p^{k - 1}}$ .

Full bookkeeping is in Diamond-Shurman ^{[Ch. 5 Proposition 5.2.4]} and Serre ^{[Ch. VII §5 Théorème 6]}. $□$

Proposition (Euler-product factorisation of $L (s, Δ)$ ). The Dirichlet series $L (s, Δ) = \sum_{n \geq 1} τ (n) n^{- s}$ factors as $\prod_{p} (1 - τ (p) p^{- s} + p^{11 - 2 s})^{- 1}$ on $Re (s) > 13/2$ .

Proof. Multiplicativity at coprime indices gives the formal Dirichlet-series factorisation $L (s, Δ) = \prod_{p} L_{p} (s)$ where $L_{p} (s) = \sum_{k \geq 0} τ (p^{k}) p^{- k s}$ . Setting $X = p^{- s}$ and $a = τ (p)$ , the recursion $τ (p^{k + 1}) = a τ (p^{k}) - p^{11} τ (p^{k - 1})$ translates into the formal identity $$ (1 - a X + p^{11} X^2) L_p(s) = \sum_{k \geq 0} \left( \tau(p^k) - a \tau(p^{k-1}) + p^{11} \tau(p^{k-2}) \right) X^k = \tau(1) + (\tau(p) - a \tau(1)) X = 1 $$ (using $τ (p^{- 1}) = 0$ and $τ (p^{- 2}) = 0$ by convention, and $τ (1) = 1$ , $τ (p) = a$ ). Hence $L_{p} (s) = (1 - τ (p) p^{- s} + p^{11 - 2 s})^{- 1}$ . Absolute convergence on $Re (s) > 13/2$ follows from the elementary bound $∣ τ (n) ∣ ≪ n^{6}$ (Hecke 1937; the sharp form $∣ τ (n) ∣ \leq d (n) n^{11/2}$ requires Deligne 1974). $□$

Proposition (functional equation $Λ (s, Δ) = Λ (12 - s, Δ)$ ). The completed $L$ -function $Λ (s, Δ) = (2 π)^{- s} Γ (s) L (s, Δ)$ extends to an entire function of $s \in C$ and satisfies $Λ (s, Δ) = Λ (12 - s, Δ)$ .

Proof. For $Re (s) > 13/2$ , write $$ \Lambda(s, \Delta) = \int_0^\infty \Delta(i y) , y^s , \frac{dy}{y}. $$ Substituting the $q$ -expansion $Δ (i y) = \sum_{n \geq 1} τ (n) e^{- 2 π n y}$ and integrating term by term (justified by absolute convergence in $Re (s) > 13/2$ ): $$ \int_0^\infty \Delta(i y) y^s \frac{dy}{y} = \sum_{n \geq 1} \tau(n) \int_0^\infty e^{-2 \pi n y} y^s \frac{dy}{y} = \sum_{n \geq 1} \tau(n) \frac{\Gamma(s)}{(2 \pi n)^s} = (2 \pi)^{-s} \Gamma(s) L(s, \Delta). $$ Split the integral at $y = 1$ and apply the weight-12 transformation $Δ (i / y) = (i y / i)^{12} Δ (i y) = y^{12} Δ (i y)$ (Exercise 7) to the integral over $(0, 1)$ : $$ \int_0^1 \Delta(i y) y^s \frac{dy}{y} \overset{y \to 1/y}{=} \int_1^\infty \Delta(i/y) y^{-s} \frac{dy}{y} = \int_1^\infty y^{12} \Delta(i y) y^{-s} \frac{dy}{y} = \int_1^\infty \Delta(i y) y^{12 - s} \frac{dy}{y}. $$ Combining, $$ \Lambda(s, \Delta) = \int_1^\infty \Delta(i y) \left( y^s + y^{12 - s} \right) \frac{dy}{y}. $$ The integrand $Δ (i y) = O (e^{- 2 π y})$ as $y \to \infty$ decays super-exponentially, so the integral converges absolutely for every $s \in C$ . The right-hand side is therefore entire in $s$ . The substitution $s \mapsto 12 - s$ exchanges the two terms $y^{s} + y^{12 - s}$ , leaving the integral invariant. Hence $Λ (s, Δ) = Λ (12 - s, Δ)$ . $□$

Proposition (Deligne 1974 — $∣ τ (p) ∣ \leq 2 p^{11/2}$ from Weil II). The Frobenius eigenvalues $α_{p}, β_{p}$ of $ρ_{Δ} (Frob_{p})$ satisfy $∣ α_{p} ∣ = ∣ β_{p} ∣ = p^{11/2}$ for every prime $p$ , and consequently $∣ τ (p) ∣ \leq 2 p^{11/2}$ .

Proof sketch. The Galois representation $ρ_{Δ}$ is realised by Deligne 1971 on the $Δ$ -isotypic component of the étale cohomology $H_{\overset{e}{ˊ} t}^{11} (E^{10}_{\overline{Q}}, Q_{ℓ})$ of the Kuga-Sato eleven-fold, the smooth proper compactification of the ten-fold fibre product of the universal elliptic curve over the modular curve $Y (1)$ . The Kuga-Sato variety has good reduction at every prime $p$ (using the smoothness of the universal elliptic curve away from the cusps and the explicit compactification by Deligne 1971). Weil II (Deligne 1974 Publ. Math. IHÉS 43) — the proof of the Riemann hypothesis for varieties over finite fields — states that for a smooth proper variety $X$ over $F_{p}$ , the eigenvalues of $Frob_{p}$ on $H_{\overset{e}{ˊ} t}^{i} (X_{\overline{F_{p}}}, Q_{ℓ})$ are algebraic integers all of whose Galois conjugates have absolute value $p^{i /2}$ . Applied to the Kuga-Sato variety with $i = 11$ , the eigenvalues of $Frob_{p}$ on $H_{\overset{e}{ˊ} t}^{11}$ have absolute value $p^{11/2}$ . The $Δ$ -isotypic component is a Galois-stable sub-quotient, so its eigenvalues are still Weil numbers of weight 11. Hence $∣ α_{p} ∣ = ∣ β_{p} ∣ = p^{11/2}$ , and $∣ τ (p) ∣ = ∣ α_{p} + β_{p} ∣ \leq ∣ α_{p} ∣ + ∣ β_{p} ∣ = 2 p^{11/2}$ . $□$

Proposition (Ramanujan congruence mod 691). For every prime $p$ , $$ \tau(p) \equiv 1 + p^{11} \pmod{691}. $$

Proof. The prime $691$ appears in the numerator of the Bernoulli number $B_{12} = - 691/2730$ . Consider the Eisenstein series of weight 12: $E_{12} (z) = 1 - (24/ B_{12}) \sum_{n \geq 1} σ_{11} (n) q^{n} = 1 + (65520/691) \sum_{n \geq 1} σ_{11} (n) q^{n}$ . The structure of $M_{12} (SL_{2} (Z))$ gives an identity $691Δ + (integer-coefficient combination of E_{4}, E_{6}) \cdot E_{12}$ with integer coefficients, from which one extracts: writing $E_{12} = 1 + (65520/691) \sum σ_{11} (n) q^{n}$ and clearing the denominator, $691 E_{12}$ has integer coefficients with $691 a_{n} (E_{12}) = 65520 σ_{11} (n)$ for $n \geq 1$ . The Hecke eigenform $Δ$ and the Hecke eigenform $E_{12}$ (the Eisenstein series is automatically a Hecke eigenform with eigenvalue $1 + p^{11}$ at $T_{p}$ , since $T_{p}$ acts on Eisenstein series via the divisor-sum identity) have eigenvalues $τ (p)$ and $1 + p^{11}$ at $T_{p}$ respectively. The congruence $691 (Δ - some scalar \cdot E_{12}) \equiv 0 (mod 691)$ at the level of Fourier coefficients descends to $τ (p) \equiv 1 + p^{11} (mod 691)$ for every prime $p$ .

The modern interpretation: the residual representation $\overset{ρ}{ˉ}_{Δ, 691} : G_{Q} \to GL_{2} (F_{691})$ is reducible, decomposing as $1 \oplus χ_{cyc}^{11}$ where $χ_{cyc} : G_{Q} \to F_{691}^{\times}$ is the mod-691 cyclotomic character. The trace of the diagonal representation at $Frob_{p}$ is $1 + p^{11} mod 691$ , matching $τ (p) mod 691$ by Eichler-Shimura. Ramanujan discovered this congruence empirically in 1916; the modular-Galois-representation explanation is Serre 1968 + Swinnerton-Dyer 1973. $□$

Connections Master

Modular forms on $SL_{2} (Z)$ 21.04.01. The modular discriminant $Δ$ is the foundational example of a cusp form on $SL_{2} (Z)$ : it is the smallest nonzero element of the graded ring $S_{*} (SL_{2} (Z))$ of cusp forms, the generator of the one-dimensional space $S_{12}$ , and the witness that $S_{k}$ is nonzero for some weight. Its Fourier coefficients $τ (n)$ are the simplest non-Eisenstein arithmetic data arising from the theory, and the entire programme of Hecke eigenforms is built around understanding them.
Hecke operators and Hecke algebra 21.04.02. The multiplicativity $τ (mn) = τ (m) τ (n)$ for coprime $(m, n)$ and the recursion $τ (p^{k + 1}) = τ (p) τ (p^{k}) - p^{11} τ (p^{k - 1})$ are direct consequences of the Hecke-algebra relations $T_{m} T_{n} = T_{mn}$ and $T_{p} T_{p^{k}} = T_{p^{k + 1}} + p^{11} T_{p^{k - 1}}$ applied to $Δ$ as a one-dimensional Hecke eigenform. The Ramanujan tau function is the prototype showing how Hecke eigenvalues control Fourier coefficients of cusp forms.
Eichler-Shimura correspondence 21.04.03. The weight-2 version of the modular-form / Galois-representation dictionary is the Eichler-Shimura correspondence; the weight-12 version applied to $Δ$ is Deligne 1971's attachment of $ρ_{Δ}$ . The trace-of-Frobenius identity $tr ρ_{Δ} (Frob_{p}) = τ (p)$ is the higher-weight analogue of the $a_{p} (f) = p + 1 - # E (F_{p})$ identity at weight 2, generalising the Eichler-Shimura congruence relation from the modular Jacobian to the Kuga-Sato variety.
$ℓ$ -adic Galois representations 21.05.01. The representation $ρ_{Δ}$ is the second-simplest example of an $ℓ$ -adic Galois representation arising from arithmetic geometry, after the Tate module of an elliptic curve. The Frobenius traces $τ (p) = tr ρ_{Δ} (Frob_{p})$ and determinants $p^{11} = det ρ_{Δ} (Frob_{p})$ provide the local data, and the absolute irreducibility / open-image properties of $ρ_{Δ}$ are the analogues for $Δ$ of Serre's open-image theorem for elliptic curves.
Riemann zeta function 21.03.01. The Dirichlet series $L (s, Δ) = \sum τ (n) n^{- s}$ is the modular-form analogue of $ζ (s) = \sum 1/ n^{s}$ . The Euler product $\prod_{p} (1 - τ (p) p^{- s} + p^{11 - 2 s})^{- 1}$ is the $GL_{2}$ analogue of the $GL_{1}$ Euler product $\prod_{p} (1 - p^{- s})^{- 1}$ ; the functional equation $Λ (s, Δ) = Λ (12 - s, Δ)$ is the analogue of $ξ (s) = ξ (1 - s)$ ; and the Ramanujan-Petersson bound $∣ τ (p) ∣ \leq 2 p^{11/2}$ is the analogue of the Riemann hypothesis prediction for the local zeros of $ζ$ .
Modularity theorem and BSD 21.06.01. The modularity theorem extends the modular-form / Galois-representation dictionary established by Deligne 1971 for $Δ$ to every elliptic curve over $Q$ : each $E$ matches a weight-2 cusp newform $f_{E}$ with $ρ_{E, ℓ} ≅ ρ_{f_{E}, ℓ}$ . Ramanujan's three conjectures for $τ$ are the first verified instance of the general programme that the modularity theorem completes for elliptic curves.

Historical & philosophical context Master

Srinivasa Ramanujan introduced the function $τ$ in his 1916 paper On certain arithmetical functions, published in the Transactions of the Cambridge Philosophical Society 22, 159-184 ^{[Ramanujan 1916]}. The paper computed the Fourier coefficients of $Δ (z) = q \prod (1 - q^{n})^{24}$ by direct expansion, tabulated $τ (n)$ for $n$ up to several dozen, and observed three patterns: multiplicativity $τ (mn) = τ (m) τ (n)$ for coprime $(m, n)$ , a two-term recursion at prime powers $τ (p^{k + 1}) = τ (p) τ (p^{k}) - p^{11} τ (p^{k - 1})$ , and the size bound $∣ τ (p) ∣ \leq 2 p^{11/2}$ at primes. Ramanujan stated all three as conjectures, supported by numerical evidence and what he called "empirical inquiry"; he gave no proofs.

Louis Mordell, then at St John's College, Cambridge, supplied the first proofs of the multiplicativity and the recursion within a year. His 1917 paper On Mr. Ramanujan's empirical expansions of modular functions (Proceedings of the Cambridge Philosophical Society 19, 117-124) ^{[Mordell 1917]} worked with the attached Dirichlet series $\sum τ (n) n^{- s}$ directly, manipulating its Mellin transform against $Δ (i y)$ and exploiting the weight-12 transformation law. Mordell did not have the language of Hecke operators — those were introduced twenty years later — but his arguments anticipated the modern framework. Erich Hecke's 1937 papers Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I, II (Mathematische Annalen 114, 1-28 and 316-351) ^{[Hecke 1937]} introduced the operators $T_{n}$ on modular forms, proved the Hecke-algebra relations, and recovered Ramanujan's multiplicativity and recursion as automatic corollaries of $Δ$ being a Hecke eigenform on a one-dimensional space — a cleaner derivation that has been the textbook approach ever since.

The size conjecture $∣ τ (p) ∣ \leq 2 p^{11/2}$ resisted attack for fifty-eight years. Hans Petersson 1939 Math. Ann. 116 generalised the conjecture to the Ramanujan-Petersson conjecture for arbitrary weight- $k$ cusp newforms: $∣ a_{p} (f) ∣ \leq 2 p^{(k - 1) /2}$ . Partial results accumulated: Rankin 1939 proved $∣ τ (n) ∣ ≪ n^{6 - 1/4}$ via the Rankin-Selberg convolution; Selberg gave similar bounds; Eichler 1954 established $∣ a_{p} ∣ \leq 2 p^{(k - 1) /2}$ for weight-2 forms via the Eichler-Shimura congruence relation on the modular Jacobian. The full conjecture for arbitrary weight $k \geq 2$ was settled by Pierre Deligne in 1971-74. The 1971 Séminaire Bourbaki exposé 355 Formes modulaires et représentations $ℓ$ -adiques ^{[Deligne 1971]} constructed the two-dimensional $ℓ$ -adic Galois representation $ρ_{f}$ attached to any normalised cusp eigenform of weight $\geq 2$ , realised on the étale cohomology of a Kuga-Sato variety, and reduced the Ramanujan-Petersson bound to the Riemann hypothesis for varieties over finite fields. The 1974 Publ. Math. IHÉS 43 La conjecture de Weil. I ^{[Deligne 1974]} proved that hypothesis (Weil II), completing the chain. Ramanujan's 1916 conjecture became a theorem fifty-eight years later, as a consequence of one of the deepest theorems in arithmetic geometry of the twentieth century.

Ramanujan's discovery of the mod-691 congruence $τ (p) \equiv 1 + p^{11} (mod 691)$ inside the same 1916 paper anticipated the entire subject of congruences between modular forms and Eisenstein series. The interpretation via the reducible residual representation $\overset{ρ}{ˉ}_{Δ, 691} ≅ 1 \oplus χ_{cyc}^{11}$ is Serre 1968 Sém. Delange-Pisot-Poitou and Swinnerton-Dyer 1973 Lecture Notes in Math. 350; their list of exceptional primes $ℓ \in {2, 3, 5, 7, 23, 691}$ for $Δ$ has been a central computational testing ground for the deformation theory of modular Galois representations developed by Mazur 1989 and used by Wiles 1995 in his proof of modularity for semistable elliptic curves. The single function $τ$ , defined by an infinite product of twenty-fourth powers, sits at the intersection of analytic number theory, modular-form theory, étale cohomology, and the Langlands programme.

Bibliography Master

@article{Ramanujan1916,
  author  = {Ramanujan, S.},
  title   = {On certain arithmetical functions},
  journal = {Transactions of the Cambridge Philosophical Society},
  volume  = {22},
  year    = {1916},
  pages   = {159--184}
}

@article{Mordell1917,
  author  = {Mordell, L. J.},
  title   = {On Mr. Ramanujan's empirical expansions of modular functions},
  journal = {Proceedings of the Cambridge Philosophical Society},
  volume  = {19},
  year    = {1917},
  pages   = {117--124}
}

@article{Hecke1937,
  author  = {Hecke, Erich},
  title   = {{\"U}ber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I, II},
  journal = {Mathematische Annalen},
  volume  = {114},
  year    = {1937},
  pages   = {1--28, 316--351}
}

@article{Petersson1939,
  author  = {Petersson, Hans},
  title   = {Konstruktion der Modulformen und der zu gewissen Grenzkreisgruppen geh{\"o}rigen automorphen Formen},
  journal = {Mathematische Annalen},
  volume  = {116},
  year    = {1939},
  pages   = {401--412}
}

@incollection{Deligne1971,
  author    = {Deligne, Pierre},
  title     = {Formes modulaires et repr{\'e}sentations $\ell$-adiques},
  booktitle = {S{\'e}minaire Bourbaki, expos{\'e} 355 (1968/69)},
  series    = {Lecture Notes in Mathematics},
  volume    = {179},
  publisher = {Springer},
  year      = {1971},
  pages     = {139--172}
}

@article{Deligne1974,
  author  = {Deligne, Pierre},
  title   = {La conjecture de Weil. I},
  journal = {Publications math{\'e}matiques de l'IH{\'E}S},
  volume  = {43},
  year    = {1974},
  pages   = {273--307}
}

@book{Serre1973,
  author    = {Serre, Jean-Pierre},
  title     = {A Course in Arithmetic},
  publisher = {Springer},
  series    = {Graduate Texts in Mathematics},
  volume    = {7},
  year      = {1973}
}

@book{Serre1968AbelianLAdic,
  author    = {Serre, Jean-Pierre},
  title     = {Abelian $\ell$-adic Representations and Elliptic Curves},
  publisher = {W. A. Benjamin},
  year      = {1968}
}

@book{DiamondShurman2005,
  author    = {Diamond, Fred and Shurman, Jerry},
  title     = {A First Course in Modular Forms},
  publisher = {Springer},
  series    = {Graduate Texts in Mathematics},
  volume    = {228},
  year      = {2005}
}

@book{ManinPanchishkin2005,
  author    = {Manin, Yuri I. and Panchishkin, Alexei A.},
  title     = {Introduction to Modern Number Theory},
  publisher = {Springer},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {49},
  edition   = {2},
  year      = {2005}
}

@book{Shimura1971,
  author    = {Shimura, Goro},
  title     = {Introduction to the Arithmetic Theory of Automorphic Functions},
  publisher = {Princeton University Press},
  series    = {Publications of the Mathematical Society of Japan},
  volume    = {11},
  year      = {1971}
}

@article{SwinnertonDyer1973,
  author    = {Swinnerton-Dyer, H. P. F.},
  title     = {On $\ell$-adic representations and congruences for coefficients of modular forms},
  journal   = {Lecture Notes in Mathematics},
  volume    = {350},
  publisher = {Springer},
  year      = {1973},
  pages     = {1--55}
}

@article{Lehmer1947,
  author  = {Lehmer, D. H.},
  title   = {The vanishing of Ramanujan's function $\tau(n)$},
  journal = {Duke Mathematical Journal},
  volume  = {14},
  year    = {1947},
  pages   = {429--433}
}

@article{Serre1981GlasgowTau,
  author  = {Serre, Jean-Pierre},
  title   = {Quelques applications du th{\'e}or{\`e}me de densit{\'e} de Chebotarev},
  journal = {Publications math{\'e}matiques de l'IH{\'E}S},
  volume  = {54},
  year    = {1981},
  pages   = {323--401}
}

@article{BLGHT2011,
  author  = {Barnet-Lamb, Thomas and Geraghty, David and Harris, Michael and Taylor, Richard},
  title   = {A family of Calabi-Yau varieties and potential automorphy II},
  journal = {Publications of the Research Institute for Mathematical Sciences},
  volume  = {47},
  year    = {2011},
  pages   = {29--98}
}

Prerequisites

21.04.01
21.04.02
21.03.01

Tier anchors

beginner: Serre *A Course in Arithmetic* (Springer GTM 7, 1973) Ch. VII §4 (the discriminant $\Delta$ and its first Fourier coefficients); Diamond-Shurman *A First Course in Modular Forms* (GTM 228, 2005) §1.2 informal opening on $\Delta$ and $\tau$
intermediate: Serre *A Course in Arithmetic* Ch. VII §§4-5 (uniqueness of $\Delta$, Hecke-operator action and the multiplicativity of $\tau$); Diamond-Shurman §§5.2-5.3 (Hecke recursion and the attached Dirichlet series); Manin-Panchishkin *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed. 2005) Ch. 6 §3 (Hecke operators on $\Delta$, Euler product for $L(s, \Delta)$)
master: Ramanujan 1916 *Trans. Cambridge Phil. Soc.* 22 (originator paper *On certain arithmetical functions*: the three conjectures on $\tau$); Mordell 1917 *Proc. Cambridge Phil. Soc.* 19 (proof of multiplicativity and the recursion); Hecke 1937 *Math. Ann.* 114 (Hecke-operator framework deriving both); Deligne 1971 *Séminaire Bourbaki* 355 (the $\ell$-adic Galois representation $\rho_\Delta$ attached to $\Delta$); Deligne 1974 *Publ. Math. IHÉS* 43 (Ramanujan-Petersson bound $|\tau(p)| \leq 2 p^{11/2}$ via Weil II); Serre 1968 *Abelian $\ell$-adic Representations and Elliptic Curves* (Benjamin); Manin-Panchishkin 2005 Ch. 6-7; Diamond-Shurman 2005 Ch. 9; Shimura 1971 *Introduction to the Arithmetic Theory of Automorphic Functions* Ch. 3; Murty *Ramanujan's Mathematics and the Tau Function* (Hardy-Ramanujan J., various)

References

Ramanujan 1916 — On certain arithmetical functions · *Transactions of the Cambridge Philosophical Society* 22, 159-184. The originator paper defining $\tau(n)$ via $\Delta = q \prod (1 - q^n)^{24} = \sum \tau(n) q^n$ and conjecturing (i) multiplicativity $\tau(mn) = \tau(m)\tau(n)$ for coprime $(m, n)$, (ii) the prime-power recursion $\tau(p^{k+1}) = \tau(p)\tau(p^k) - p^{11}\tau(p^{k-1})$, (iii) the bound $|\tau(p)| \leq 2 p^{11/2}$ for every prime $p$.
Mordell 1917 — On Mr. Ramanujan's empirical expansions of modular functions · *Proceedings of the Cambridge Philosophical Society* 19, 117-124. The first proof of conjectures (i) and (ii) by a direct calculation with the Dirichlet series $\sum \tau(n) n^{-s}$, anticipating the Hecke-operator framework by twenty years.
Hecke 1937 — Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II · *Mathematische Annalen* 114, 1-28 + 316-351. The Hecke-operator derivation of the multiplicativity and recursion as automatic consequences of $\Delta$ being a Hecke eigenform on a one-dimensional space.
Serre 1973 — A Course in Arithmetic · Springer Graduate Texts in Mathematics 7. Ch. VII §§4-5 — the cleanest textbook account of $\Delta$, its uniqueness as the weight-12 normalised cusp form on $\mathrm{SL}_2(\mathbb{Z})$, the Hecke-operator action, and the multiplicativity and recursion for $\tau$.
Deligne 1971 — Formes modulaires et représentations $\ell$-adiques · *Séminaire Bourbaki*, exposé 355 (1968/69), Lecture Notes in Math. 179, Springer (1971), 139-172. Construction of the $\ell$-adic Galois representation $\rho_\Delta : G_\mathbb{Q} \to \mathrm{GL}_2(\overline{\mathbb{Q}_\ell})$ attached to $\Delta$, with $\mathrm{tr}\,\rho_\Delta(\mathrm{Frob}_p) = \tau(p)$ and $\det \rho_\Delta(\mathrm{Frob}_p) = p^{11}$ for every $p \neq \ell$, realised on the étale cohomology of the Kuga-Sato eleven-fold.
Deligne 1974 — La conjecture de Weil. I · *Publications mathématiques de l'IHÉS* 43, 273-307. Proof of the Riemann hypothesis for varieties over finite fields; the Ramanujan-Petersson bound $|\tau(p)| \leq 2 p^{11/2}$ follows by applying purity of the Frobenius eigenvalues on the étale cohomology of the Kuga-Sato variety to $\rho_\Delta$.
Diamond-Shurman 2005 — A First Course in Modular Forms · Springer Graduate Texts in Mathematics 228. Ch. 1 §1.2 (the discriminant $\Delta$); Ch. 5 §§5.2-5.3 (Hecke operators on cusp forms, multiplicativity, the Dirichlet series and its Euler product); Ch. 9 (Galois representations attached to modular forms, Deligne 1971).
Manin-Panchishkin 2005 — Introduction to Modern Number Theory · Springer Encyclopaedia of Mathematical Sciences 49, 2nd edition. Ch. 6 §3 (Hecke operators, $\Delta$ as the prototypical eigenform); Ch. 6 §4 (the $L$-function $L(s, \Delta)$, functional equation, Euler product); Ch. 7 (Galois representations, Ramanujan-Petersson via Weil II).
Shimura 1971 — Introduction to the Arithmetic Theory of Automorphic Functions · Publications of the Mathematical Society of Japan 11, Princeton University Press. Ch. 3 (Hecke operators on cusp forms, eigenform-coefficient identity, multiplicativity); the canonical advanced reference.
Serre 1968 — Abelian $\ell$-adic Representations and Elliptic Curves · W. A. Benjamin, New York. The founding monograph for the $\ell$-adic Galois-representation formalism; positions $\rho_\Delta$ as the analogue for $\Delta$ of the Tate module of an elliptic curve.

Estimated time

beginner: 20m
intermediate: 50m
master: 90m