Ramanujan tau function

The Ramanujan tau function, denoted \tau(n), is a multiplicative arithmetic function of a positive integer n defined by the Fourier expansion of the modular discriminant \Delta(z) = \sum_{n=1}^\infty \tau(n) q^n = q \prod_{k=1}^\infty (1 - q^k)^{24}, where q = e^{2\pi i z} and \Delta(z) is a cusp form of weight 12 for the modular group \mathrm{SL}_2(\mathbb{Z}).^[1] Introduced by the Indian mathematician Srinivasa Ramanujan in 1916, \tau(n) first appeared as the coefficient in an "error term" for the number of representations of n as a sum of 24 squares, arising in his investigations of generalized divisor functions and identities involving sums of powers.^[1] Ramanujan observed that \tau(n) satisfies the multiplicativity property \tau(mn) = \tau(m) \tau(n) whenever m and n are coprime, a relation he conjectured based on empirical computations of its values; this was rigorously proved by Louis Joel Mordell in 1917 using properties of modular functions and Dirichlet series.^[2] The first few values are \tau(1) = 1, \tau(2) = -24, \tau(3) = 252, \tau(4) = -1472, \tau(5) = 4830, illustrating its alternating signs and rapid growth.^[3] Ramanujan also conjectured a precise bound on its growth at primes, |\tau(p)| \leq 2p^{11/2} for prime p, which implies |\tau(n)| = O(n^{11/2 + \epsilon}) for any \epsilon > 0; this bound, known as the Ramanujan–Petersson conjecture in the broader context of Hecke eigenforms, was established by Pierre Deligne in 1974 as a consequence of his proof of the Weil conjectures.^[4] Beyond its arithmetic properties, the tau function holds profound significance in number theory and algebraic geometry, serving as the prototype for coefficients of normalized Hecke eigenforms and appearing in identities linking partitions, elliptic curves, and L-functions.^[5] Ramanujan proposed several congruences, such as \tau(p) \equiv \sigma_{11}(p) \pmod{691} for primes p, many of which were verified shortly after and relate to the non-vanishing of \tau(n), as conjectured by Derrick Henry Lehmer in 1941 (still open).^[1] Its study has influenced modern developments in automorphic forms, trace formulas, and the Langlands program, underscoring Ramanujan's intuitive grasp of deep analytic structures.

Definition and Context

Mathematical definition

The Ramanujan tau function arises in the theory of modular forms, which are holomorphic functions on the upper half-plane \mathbb{H} = \{ [z](/page/Z) \in \mathbb{C} \mid \Im(z) > 0 \} that transform in a specific way under the action of the modular group \mathrm{SL}(2, \mathbb{Z}), and admit Fourier expansions of the form \sum_{n=0}^\infty a_n q^n with q = e^{2\pi i [z](/page/Z)}.^[6] The tau function \tau(n) is defined for positive integers n as the sequence of coefficients in the q-expansion of the modular discriminant:

\sum_{n=1}^\infty \tau(n) q^n = q \prod_{m=1}^\infty (1 - q^m)^{24},

where q = e^{2\pi i [z](/page/Z)} with [z](/page/Z) \in \mathbb{H}.^[1]^[6] This infinite product equals \eta(z)^{24}, where \eta(z) is the Dedekind eta function, given by \eta(z) = q^{1/24} \prod_{m=1}^\infty (1 - q^m). The function \Delta(z) := \eta(z)^{24} is thus the unique normalized cusp form of weight 12 for \mathrm{SL}(2, \mathbb{Z}), meaning it is holomorphic on \mathbb{H}, vanishes at the cusps, and satisfies the transformation law f\left( \frac{az + b}{cz + d} \right) = (cz + d)^{12} f(z) for all \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2, \mathbb{Z}).^[7]^[6]

Historical background

The Ramanujan tau function, denoted \tau(n), was introduced by the Indian mathematician Srinivasa Ramanujan in his 1916 paper "On certain arithmetical functions," where it emerged as the Fourier coefficients of the discriminant modular form \Delta(z) = q \prod_{n=1}^\infty (1 - q^n)^{24}, with q = e^{2\pi i z}. This discovery was motivated by Ramanujan's investigations into partition identities and the arithmetic of sums of squares, particularly as an error term in counting representations of integers by 24 squares. The function's definition via this q-expansion highlighted its deep connections to elliptic functions and theta series, though Ramanujan provided empirical observations rather than full theoretical foundations. Ramanujan's initial insights into the properties of \tau(n) appeared during his correspondence with G.H. Hardy between 1913 and 1914, where he alluded to its multiplicative behavior and growth estimates without providing complete proofs, prompting Hardy to invite him to Cambridge for collaborative work. Following publication, Louis Mordell established the multiplicativity of \tau(n) in 1917, showing that \tau(mn) = \tau(m)\tau(n) for coprime m and n using properties of modular forms. Ramanujan had conjectured a Hecke bound |\tau(p)| \leq 2p^{11/2} for primes p, reflecting the function's bounded growth; this was later rigorously proved by Pierre Deligne in 1974 through the machinery of étale cohomology, confirming the Ramanujan-Petersson conjecture. In the 1930s, computational efforts advanced the study of \tau(n), with Derrick Henry Lehmer calculating values up to n = 1040 using early mechanical aids, which verified Ramanujan's tabulated values and supported conjectures on non-vanishing and sign patterns. These computations underscored the function's irregularity while aligning with modular form theory. By the mid-20th century, \tau(n) was recognized as the coefficients of the unique normalized cusp form of weight 12 on the full modular group \mathrm{[SL](/page/SL)}(2, \mathbb{Z}), a result stemming from the dimension formula for the space of cusp forms, established through Erich Hecke's development of Hecke operators in the 1930s.

Basic Properties

Computed values

The values of the Ramanujan tau function for $1 \leq n \leq 20 are listed in the table below.^[3]

n	\tau(n)
1	1
2	-24
3	252
4	-1472
5	4830
6	-6048
7	-16744
8	84480
9	-113643
10	-115920
11	534612
12	-370944
13	-577738
14	401856
15	1217160
16	987136
17	-6905934
18	2727432
19	10661420
20	-7109760

A notable pattern among these values is that \tau(n) is odd if and only if n is an odd square; it is even otherwise.^[8] These initial values indicate that |\tau(n)| grows roughly on the order of n^6, consistent with the weight 12 of the associated cusp form \Delta(z).^[4] For small n, \tau(n) can be computed by expanding the defining product formula

\Delta(z) = q \prod_{m=1}^\infty (1 - q^m)^{24} = \sum_{n=1}^\infty \tau(n) q^n,

truncating the infinite product at a suitable point to obtain the coefficient of q^n.^[1]

Multiplicativity and recurrence

The Ramanujan τ function exhibits multiplicativity, satisfying τ(mn) = τ(m) τ(n) whenever m and n are coprime positive integers. This arithmetic property was conjectured by Ramanujan based on empirical observations of the function's values and rigorously proved by Mordell in 1917 through an analysis of the associated modular form and its transformation properties under the modular group.^[2] The multiplicativity of τ(n) arises from the Euler product structure of its associated Dirichlet L-function, defined as
L(s) = \sum_{n=1}^\infty \frac{\tau(n)}{n^s} = \prod_p \left(1 - \frac{\tau(p)}{p^s} + \frac{p^{11}}{p^{2s}}\right)^{-1},
where the product runs over all primes p; this factorization encodes the behavior at each prime and directly implies the multiplicative relation for coprime arguments.^[9] Mordell's proof simultaneously establishes a linear recurrence relation governing the values of τ at prime powers. Specifically, for any prime p and integer r ≥ 1,
\tau(p^{r+1}) = \tau(p) \, \tau(p^r) - p^{11} \, \tau(p^{r-1}),
with the base case τ(p^0) = τ(1) = 1. This relation allows iterative computation of τ(p^k) starting from τ(p), reflecting the local structure captured in the Euler factor for each p.^[2] Together, multiplicativity and the prime power recurrence enable efficient determination of τ(n) for arbitrary n via its prime factorization: if n = \prod_p p^{k_p}, then τ(n) = \prod_p τ(p^{k_p}), where each τ(p^{k_p}) is obtained recursively. This algebraic framework underpins both theoretical analysis and numerical evaluation of the function, as illustrated by its application to small composites like τ(6) using values at 2 and 3.^[9]

Conjectures

Ramanujan's original conjectures

In 1916, Srinivasa Ramanujan introduced the tau function τ(n) as the coefficients in the q-expansion of the modular form Δ(q) = η(q)^{24}, where η(q) is the Dedekind eta function, and proposed three key conjectures based on his computations of the first several values. These conjectures emerged from his investigations into arithmetic properties linked to partition congruences, such as the famous result that the number of partitions p(5n + 4) is divisible by 5, which Ramanujan connected to the behavior of Δ(q).^[1] Ramanujan first conjectured the multiplicativity of τ(n), stating that τ(mn) = τ(m)τ(n) whenever m and n are coprime positive integers. This property allows τ(n) to be determined entirely from its values at prime powers, facilitating computational and analytic studies. Louis J. Mordell proved this conjecture in 1917 using properties of modular forms and Eisenstein series.^[1]^[10] Ramanujan also conjectured a specific recurrence relation for τ(p^k) at prime powers p^k, expressing τ(p^k) in terms of τ(p) via a trigonometric formula: defining θ_p such that cos θ_p = τ(p) / (2 p^{11/2}), then p^{-11k/2} τ(p^k) = sin((k+1) θ_p) / sin(θ_p). This relation, which generalizes to the Hecke operator action on cusp forms, was similarly established by Mordell in 1917 through verification of the modular transformation properties of Δ(q).^[1]^[10] The most celebrated of Ramanujan's conjectures is the bound |τ(p)| ≤ 2 p^{11/2} for prime p, which implies that the Fourier coefficients grow no faster than expected for a weight-12 cusp form and has profound implications for the associated L-function. This remained unproven for nearly six decades until Pierre Deligne established it in 1974 as a consequence of his proof of the Weil conjectures, using étale cohomology on the moduli stack of elliptic curves.^[1]^[4]

Modern conjectures

Lehmer's conjecture, proposed by Derrick Henry Lehmer in 1947, asserts that the Ramanujan tau function satisfies \tau(n) \neq 0 for all positive integers n \geq 1. This remains unproven. Computational verifications have confirmed the conjecture for all n < 8.162 \times 10^{23}, using algorithms based on computing Galois representations modulo primes to establish non-vanishing in arithmetic progressions. A related open question concerns the signs of \tau(n), with the conjecture that the sign changes infinitely often, implying infinitely many positive and negative values. This has been established rigorously, showing that \tau(n) takes both positive and negative values infinitely often, with explicit bounds on the frequency of sign changes in short intervals. These results align with probabilistic models inspired by Cohen-Lenstra heuristics on the distribution of arithmetic invariants, though direct applications remain exploratory. Conjectures on prime values of \tau(n) include the expectation that |\tau(n)| is prime for infinitely many n, motivated by the irregular distribution of the function's values and analogies with other arithmetic sequences. Specifically for primes p, it is anticipated that \tau(p) takes prime values infinitely often, though no such instances are known beyond small p due to the rapid growth of |\tau(p)| \approx 2 p^{11/2}. Recent progress indicates that prime values of \tau(n) form thin sets in most residue classes modulo 23, with positive density zero in those classes, limiting their arithmetic distribution.^[11] The Sato-Tate conjecture provides a precise description of the distribution of normalized prime values \tau(p) / (2 p^{11/2}) for odd primes p. It predicts that these normalized coefficients are equidistributed with respect to the Sato-Tate measure \frac{2}{\pi} \sqrt{1 - x^2} \, dx on [-2, 2], reflecting the Frobenius conjugacy classes in the associated Galois representation. This was proved by Pierre Deligne in 1974 as a consequence of his solution to the Weil conjectures for the modular curve associated to the discriminant modular form.

Congruences and Formulas

Key congruences

Ramanujan observed that for prime numbers p, the tau function satisfies the congruence \tau(p) \equiv \sigma_{11}(p) \pmod{691}, where \sigma_{11}(p) = 1 + p^{11} denotes the sum of the 11th powers of the divisors of p.^[1] This observation highlighted a deep connection between the tau function and divisor sums, arising from the modular form properties of the discriminant \Delta. Ramanujan established several fundamental congruences for \tau(n) modulo the primes 2, 3, 5, 7, and 23. In particular, \tau(n) \equiv 0 \pmod{2} for all n > 1. He also proved congruences modulo 3, 5, 7, and 23, including the condition that \tau(p) \equiv 0 \pmod{23} for primes p = 23 or \left( \frac{p}{23} \right) = -1. For example, \tau(n) \equiv n \sigma_3(n) \pmod{7} under certain conditions on n modulo 7. These results were derived using the multiplicative nature of \tau(n) and explicit computations for small primes, aiding later investigations, such as Lehmer's 1941 conjecture on the non-vanishing of \tau(n). Serre's theorem from the 1970s extended Ramanujan's observation to all positive integers n, proving \tau(n) \equiv \sigma_{11}(n) \pmod{691} holds universally. This result relies on the theory of modular forms modulo primes and \ell-adic representations. Serre further generalized the phenomenon to other primes dividing the numerator of Bernoulli numbers B_k for even k \leq 12, such as 2, 3, 5, 7, 23, and 691, where the Eisenstein series E_{12} is congruent to 1 modulo the prime, leading to analogous divisor sum congruences for \tau(n).

Explicit formulas

One explicit expression for the Ramanujan tau function arises from its multiplicativity and the fact that it is an eigenfunction of the Hecke operators. For a prime power p^k, the values satisfy the linear recurrence relation

\tau(p^k) = \tau(p) \tau(p^{k-1}) - p^{11} \tau(p^{k-2}),

with initial conditions \tau(p^0) = 1 and \tau(p^1) = \tau(p). This relation follows from the action of the Hecke operator T_p on the cusp form \Delta(z), where \tau(p) is the eigenvalue. Since \tau(n) is multiplicative, for n = \prod p_i^{k_i} with distinct primes p_i, it is given by \tau(n) = \prod \tau(p_i^{k_i}), allowing computation via the prime power recurrences. In 1975, Douglas Niebur derived an elementary closed-form expression for \tau(n) in terms of the divisor sum function \sigma(m) = \sum_{d \mid m} d. The formula is \begin{align*} \tau(n) &= n^4 \sigma(n) - 24 \sum_{k=1}^{n-1} \left(35 k^4 - 52 n k^3 + 18 n^2 k^2 \right) \sigma(k) \sigma(n-k). \end{align*} ^[12] This sum provides a direct way to compute \tau(n) using only divisor sums up to n, without relying on modular form theory. Niebur obtained it by expressing the modular discriminant \Delta(z) via its logarithmic derivative and integrating against Eisenstein series. A more recent development provides a recursive formula linking \tau(n) directly to the ordinary divisor sum \sigma(n). In 2023, López-Bonilla et al. established the relation

n \tau(n+1) = -24 \sum_{j=1}^n \sigma(j) \tau(n+1 - j), \quad n \geq 1,

with \tau(1) = 1.^[13] Solving this recurrence yields explicit expressions for \tau(n), such as representations involving Bell polynomials or compositions of n, offering an alternative computational path. For an analytic perspective, the Petersson trace formula yields a series representation involving Kloosterman sums and Bessel functions:

\tau(n) = \frac{2^{23} \pi^{12} n^{11/2}}{ \|\Delta\|^2 \cdot 10! } \sum_{c=1}^\infty \frac{S(1,n;c)}{c} J_{11} \left( 4\pi \sqrt{\frac{n}{c}} \right), \quad n > 1,

where S(1,n;c) is the Kloosterman sum \sum_{(d,c)=1} e^{2\pi i (d^{-1} + d n)/c} and J_{11} is the Bessel function of the first kind of order 11. This converges rapidly for large n and stems from the orthogonality of cusp forms in the space of weight 12 modular forms, which contains only \Delta(z).

Analytic Aspects

Associated L-function

The associated L-function of the Ramanujan tau function \tau(n) is defined by the Dirichlet series

L(s, \tau) = \sum_{n=1}^\infty \frac{\tau(n)}{n^s},

which converges absolutely in the half-plane \Re(s) > 13/2, arising from the growth bound |\tau(n)| \ll n^{11/2 + \epsilon} for any \epsilon > 0.^[14] Since \tau(n) is a multiplicative function, this L-function admits an Euler product representation

L(s, \tau) = \prod_p \left( 1 - \tau(p) p^{-s} + p^{11 - 2s} \right)^{-1},

where the product runs over all primes p.^[14] This L-function is attached to the cusp form \Delta(z) = q \prod_{n=1}^\infty (1 - q^n)^{24} = \sum_{n=1}^\infty \tau(n) q^n, the unique normalized newform of weight 12 and level 1 for the modular group \mathrm{SL}_2(\mathbb{Z}). It satisfies the functional equation

(2\pi)^{-s} \Gamma(s) L(s, \tau) = (2\pi)^{s-12} \Gamma(12 - s) L(12 - s, \tau),

which is symmetric about the critical line \Re(s) = 6.^[14] The completed L-function is given by

\Lambda(s, \tau) = (2\pi)^{-s} \Gamma(s) L(s, \tau),

and the functional equation can equivalently be expressed as \Lambda(s, \tau) = \Lambda(12 - s, \tau).^[14] As the L-function of a cusp form of even integral weight, L(s, \tau) admits an analytic continuation to a holomorphic function on the entire complex plane, with no poles. This entire holomorphy follows from the general theory of L-functions associated to modular forms developed by Hecke.^[14]

Distribution and non-vanishing

The Ramanujan tau function \tau(n) plays a prominent role in the exact formula for r_{24}(n), the number of ways to represent a positive integer n as a sum of 24 squares, where it appears as the principal "error term" correcting the main arithmetic contribution from a weighted divisor sum. This formula, discovered by Ramanujan, highlights \tau(n) as the key modular correction ensuring the exact count of representations, with applications in analytic number theory for estimating lattice point discrepancies in higher dimensions.^[15] Lehmer's conjecture posits that \tau(n) \neq 0 for all positive integers n, a longstanding open problem first stated in 1947. Computational verifications have confirmed this non-vanishing up to extremely large bounds, such as n \leq 2.279 \times 10^{19} as established by Bosman in 2014, with no counterexamples found in subsequent checks extending into the 2020s. The conjecture has significant implications for the associated L-function L(s, \Delta) = \sum_{n=1}^\infty \tau(n) n^{-s}, as a vanishing \tau(n) would impose constraints on the location and multiplicity of its zeros, particularly ruling out certain exceptional real zeros near the critical line that could affect prime number theorems in arithmetic progressions.^[16]^[17] The asymptotic distribution of \tau(p) for primes p is governed by the Sato-Tate conjecture, which describes the normalized values \tau(p)/(2 p^{11/2}) = \cos \theta_p as following the Sato-Tate measure \frac{2}{\pi} \sin^2 \theta \, d\theta on [0, \pi]. This equidistribution result, specific to the Ramanujan \Delta-modular form, was proved by Deligne in 1974 as part of his resolution of the Weil conjectures, confirming the conjectured statistical behavior and bounding the growth of \tau(p). The proof relies on the étale cohomology of the modular curve and establishes that the angles \theta_p are asymptotically uniformly distributed with respect to this measure.^[18] Recent advances have explored the rarity of prime values of \tau(n) within arithmetic progressions. In particular, for most residue classes b \pmod{23} (specifically 16 out of 22 nonzero classes), the set of primes p \equiv b \pmod{23} that appear as \tau(n) for some n forms a thin set, meaning its counting function grows slower than any positive power of the input size, with Dirichlet density zero. This 2023 result leverages modular congruences and effective versions of the Sato-Tate distribution to show that prime \tau-values avoid most progressions modulo 23.^[11] Studies on even values of \tau(n) have also progressed, addressing multiplicity and surjectivity questions in the spirit of Lehmer's non-vanishing. For primes \ell with $3 \leq \ell < 100, \tau(n) avoids infinitely many powers \pm 2 \ell^j in specific arithmetic progressions modulo 44, including all linear, quadratic, and most cubic powers; for example, \tau(n) \neq \pm 2 \cdot 97^j unless j \equiv 0 \pmod{44}. These 2021 findings demonstrate that even integers of certain forms are not in the image of \tau, with multiplicity bounded in targeted families, using Hecke relations and distribution results.^[19]