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A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).
(Formerly M5153 N2237)
1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920, 534612, -370944, -577738, 401856, 1217160, 987136, -6905934, 2727432, 10661420, -7109760, -4219488, -12830688, 18643272, 21288960, -25499225, 13865712, -73279080, 24647168 (list; graph; refs; listen; history; text; internal format)



Coefficients of the cusp form of weight 12 for the full modular group.

It is conjectured that tau(n) is never zero (this has been verified for n < 816212624008487344127999, see the Derickx, van Hoeij, Zeng reference).

M. J. Hopkins mentions that the only known primes p for which tau(p) == 1 mod p are 11, 23 and 691, that it is an open problem to decide if there are infinitely many such p and that no others are known below 35000. Simon Plouffe has now searched up to tau(314747) and found no other examples. - N. J. A. Sloane, Mar 25 2007

Number 1 of the 74 eta-quotients listed in Table I of Martin 1996.

With Dedekind's eta function and the discriminant Delta one has eta(z)^24 = Delta(z)/(2*Pi)^12 = Sum_{m >= 1} tau(m)*q^m, with q = exp(2*Pi*i*z), and z in the complex upper half plane, where i is the imaginary unit. Delta is the eigenfunction of the Hecke operator T_n (n >= 1) with eigenvalue tau(n): T_n Delta = tau(n) Delta. From this the formula for tau(m)*tau(n) given below in the formula section follows. See, e.g., the Koecher-Krieg reference, Lemma and Satz, p. 212. Or the Apostol reference, eq. (3) on p. 114 and the first part of section 6.13 on p. 131. - Wolfdieter Lang, Jan 26 2016

For the functional equation satisfied by the Dirichlet series F(s), Re(s) > 7, of a(n) see the Hardy reference, p. 173, (10.9.4). It is (2*Pi)^(-s) * Gamma(s) * F(s) = (2*Pi)^(s-12) * Gamma(12-s) * F(12-s). This is attributed to J. R. Wilton, 1929, on p. 185. - Wolfdieter Lang, Feb 08 2017


Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 114, 131.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Farkas and Kra, Theta constants, Riemann surfaces and the modular group, AMS 2001; see p. 298.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.2).

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, lecture X, pp. 161-185.

M. J. Hopkins, Algebraic topology and modular forms, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317.

Bruce Jordan and Blair Kelly (blair.kelly(AT)att.net), The vanishing of the Ramanujan tau function, preprint, 2001.

Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 210 - 212.

M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.

N. Laptyeva, V. K. Murty, Fourier coefficients of forms of CM-type, Indian Journal of Pure and Applied Mathematics, October 2014, Volume 45, Issue 5, pp 747-758

Yu. I. Manin, Mathematics and Physics, Birkhaeuser, Boston, 1981.

H. McKean and V. Moll. Elliptic Curves, Camb. Univ. Press, p. 139.

M. Ram Murty, The Ramanujan tau-function, pp. 269-288 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.

S. Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 153, Ed. G. H. Hardy et al., AMS Chelsea 2000.

S. Ramanujan, On Certain Arithmetical Functions. Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.

J.-P. Serre, A course in Arithmetic, Springer-Verlag, 1973, see p. 98.

J.-P. Serre, Sur la lacunarité des puissances de eta, Glasgow Math. Journal, 27 (1985), 203-221.

J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 482.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. P. F. Swinnerton-Dyer, Congruence properties of tau(n), pp. 289-311 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.

D. Zagier, Introduction to Modular Forms, Chapter 4 in M. Waldschmidt et al., editors, From Number Theory to Physics, Springer-Verlag, 1992.

Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103.


Simon Plouffe, Table of n, a(n) for n = 1..16090

B. C. Berndt and K. Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary

B. C. Berndt and K. Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)

F. Brunault, La fonction Tau de Ramanujan

D. X. Charles, Computing The Ramanujan Tau Function

B. Cloitre, On the fractal behavior of primes, 2011.

John Cremona, Home page

Maarten Derickx, Mark van Hoeij, Jinxiang Zeng, Computing Galois representations and equations for modular curves X_H(l), arXiv:1312.6819 [math.NT], (18-March-2014)

B. Edixhoven et al., Computing the coefficients of a modular form, arXiv:math/0605244 [math.NT], 2006-2009.

J. A. Ewell, Ramanujan's Tau Function, Proc. Amer. Math. Soc. 128 (2000), 723-726.

J. A. Ewell, Ramanujan's Tau Function

S. R. Finch, Modular forms on SL_2(Z)

M. Z. Garaev, V. C. Garcia and S. V. Konyagin, Waring problem with the Ramanujan tau function, arXiv:math/0607169 [math.NT], 2006.

H. Gupta, The Vanishing of Ramanujan's Function(n), Current Science, 17 (1948), p. 180.

J. L. Hafner and J. Stopple, A Heat Kernel Associated to Ramanujan's Tau Function, The Ramanujan Journal 4(2) 2000,

Yang-Hui He and John McKay, Moonshine and the Meaning of Life, arXiv:1408.2083 [math.NT], 2014.

Jerry B. Keiper, Ramanujan's Tau-Dirichlet Series

D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433.

D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]

D. H. Lehmer, Tables of Ramanujan's function tau(n), Math. Comp., 24 (1970), 495-496.

LMFDB, Newform 1.12.1.a

F. Luca and I. E. Shparlinski, Arithmetic properties of the Ramanujan function, arXiv:math/0607591 [math.NT], 2006.

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

K. Matthews, Computing Ramanujan's tau function

S. C. Milne, New infinite families of exact sums of squares formulas, Jacobi elliptic functions and Ramanujan's tau function, Proc. Nat. Acad. Sci. USA, 93 (1996) 15004-15008.

S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.

Louis J. Mordell, On Mr. Ramanujan's empirical expansions of modular functions, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.

P. Moree, On some claims in Ramanujan's 'unpublished' manuscript on the partition and tau functions, arXiv:math/0201265 [math.NT], 2002.

M. R. Murty, V. K. Murty, The Ramanujan tau-function, in: The mathematical legacy of Srinivasa Ramanujan (Springer, 2012), p 11-23

M. R. Murty, V. K. Murty and T. N. Shorey, Odd values of the Ramanujan tau-function

Douglas Niebur, A formula for Ramanujan's tau-function, Illinois Journal of Mathematics, vol.19, no.3, pp.448-449, (1975). - Joerg Arndt, Sep 06 2015

Oklahoma State Mathematics Department, Ramanujan tau L-Function

J. Perry, Ramanujan's Tau Function (broken link?)

Simon Plouffe, The first 225035 terms

S. Ramanujan, Collected Papers, Table of tau(n);n=1 to 30

J. P. Serre, An interpretation of some congruences concerning Ramanujan's tau function

J. P. Serre, An interpretation of some congruences concerning Ramanujan's Tau function

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, My Favorite Integer Sequences, arXiv:math/0207175 [math.CO], 2002.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

D. A. Steffen, Les Coefficients de Fourier de la forme modulaire: La fonction de Ramanujan tau(n)

William Stein, Database

H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

G. N. Watson, A table of Ramanujan's function tau(n), Proc. London Math. Soc., 51 (1950), 1-13.

Eric Weisstein's World of Mathematics, Tau Function

K. S. Williams, Historical remark on Ramanujan's tau function, Amer. Math. Monthly, 122 (2015), 30-35.

Index entries for "core" sequences

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Index entries for sequences related to Chebyshev polynomials.


G.f.: x * Product_{k>=1} (1 - x^k)^24 = x*A(x)^8, with the g.f. of A010816.

G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 04 2011

abs(a(n)) = O(n^(11/2 + epsilon)), abs(a(p)) <= 2 p^(11/2) if p is prime. These were conjectured by Ramanujan and proved by Deligne.

Zagier says: The proof of these formulas, if written out from scratch, has been estimated at 2000 pages; in his book Manin cites this as a probable record for the ratio: "length of proof:length of statement" in the whole of mathematics.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u + 48*v + 4096*w) - v^3. - Michael Somos, Jul 19 2004

G.f. A(q) satisfies q * d log(A(q))/dq = A006352(q). - Michael Somos, Dec 09 2013

a(2*n) = A099060(n). a(2*n + 1) = A099059(n). - Michael Somos, Apr 17 2015

a(n) = tau(n) (with tau(0) = 0): tau(m)*tau(n) = Sum_{d| gcd(m,n)} d^11*tau(m*n/d^2), for positive integers m and n. If gcd(m,n) = 1 this gives the multiplicativity of tau. See a comment above with the Koecher-Krieg reference, p. 212, eq. (5). - Wolfdieter Lang, Jan 21 2016

Dirichlet series as product: Sum_{n >= 1} a(n)/n^s = Prod_{n >= 1} 1/(1 - a(prime(n))/prime(n)^s  + prime(n)^(11-2*s)). See the Mordell link, eq. (2). - Wolfdieter Lang, May 06 2016. See also Hardy, p. 164, eqs. (10.3.1) and (10.3.8). - Wolfdieter Lang, Jan 27 2017

a(n) is multiplicative with a(prime(n)^k) = sqrt(prime(n)^(11))^k*S(k, a(n) / sqrt(prime(n)^(11))), with the Chebyshev S polynomials (A049310), for n >= 1 and k >= 2, and A076847(n) = a(prime(n))). See A076847 for alpha multiplicativity and examples. - Wolfdieter Lang, May 17 2016. See also Hardy, p. 164, eq. (10.3.6) rewritten in terms of S. Wolfdieter Lang, Jan 27 2017

G.f. eta(z)^24 (with q = exp(2*Pi*i*z)) also (E_4(q)^3 - E_6(q)^2) / 1728. See the Hardy reference, p. 166, eq. (10.5.3), with Q = E_4 and R = E_6, given in A004009 and A013973, respectively. - Wolfdieter Lang, Jan 30 2017

a(n) (mod 5) == A126832(n).


G.f. = q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 + ...

35328 = (-24)*(-1472) = a(2)*a(4) = a(2*4) + 2^11*a(2*4/4) = 84480 + 2048*(-24) = 35328. See a comment on T_n Delta = tau(n) Delta above. - Wolfdieter Lang, Jan 21 2016


M := 50; t1 := series(x*mul((1-x^k)^24, k=1..M), x, M); A000594 := n-> coeff(t1, x, n);


CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 30} ]], 30], x] (* Or *)

(* first do *) Needs["NumberTheory`Ramanujan`"] (* then *) Table[ RamanujanTau[n], {n, 30}] (* Dean Hickerson, Jan 03 2003 *)

max = 28; g[k_] := -BernoulliB[k]/(2k) + Sum[ DivisorSigma[k - 1, n - 1]*q^(n - 1), {n, 2, max + 1}]; CoefficientList[ Series[ 8000*g[4]^3 - 147*g[6]^2, {q, 0, max}], q] // Rest (* Jean-François Alcover, Oct 10 2012, from modular forms *)

RamanujanTau[Range[40]] (* The function RamanujanTau is now part of Mathematica's core language so there is no longer any need to load NumberTheory`Ramanujan` before using it *) (* Harvey P. Dale, Oct 12 2012 *)

a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^24, {q, 0, n}]; (* Michael Somos, May 27 2014 *)

a[ n_] := With[{t = Log[q] / (2 Pi I)}, SeriesCoefficient[ Series[ DedekindEta[t]^24, {q, 0, n}], {q, 0, n}]]; (* Michael Somos, May 27 2014 *)


(MAGMA) M12:=ModularForms(Gamma0(1), 12); t1:=Basis(M12)[2]; PowerSeries(t1[1], 100); Coefficients($1);

(MAGMA) Basis( CuspForms( Gamma1(1), 12), 100)[1]; /* Michael Somos, May 27 2014 */

(PARI) {a(n) = if( n<1, 0, polcoeff( x * eta(x + x * O(x^n))^24, n))};

(PARI) {a(n) = if( n<1, 0, polcoeff( x * (sum( i=1, (sqrtint( 8*n - 7) + 1) \ 2, (-1)^i * (2*i - 1) * x^((i^2 - i)/2), O(x^n)))^8, n))};

(PARI) taup(p, e)={


        (65*sigma(p, 11)+691*sigma(p, 5)-691*252*sum(k=1, p-1, sigma(k, 5)*sigma(p-k, 5)))/756


        my(t=taup(p, 1));

        sum(j=0, e\2,

            (-1)^j*binomial(e-j, e-2*j)*p^(11*j)*t^(e-2*j)




a(n)=my(f=factor(n)); prod(i=1, #f[, 1], taup(f[i, 1], f[i, 2]));

\\ Charles R Greathouse IV, Apr 22 2013

(PARI) \\ compute terms individually (Douglas Niebur, Ill. J. Math., 19, 1975):

a(n) = n^4*sigma(n) - 24*sum(k=1, n-1, (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k));

vector(33, n, a(n)) \\ Joerg Arndt, Sep 06 2015

(PARI) a(n)=ramanujantau(n) \\ Charles R Greathouse IV, May 27 2016

(Sage) CuspForms( Gamma1(1), 12, prec=100).0; # Michael Somos, May 28 2013

(Sage) list(delta_qexp(100))[1:] # faster Peter Luschny, May 16 2016


Cf. A076847 (tau(prime)), A278577 (prime powers), A037955, A027364, A037945, A037946, A037947, A008408 (Leech).

For a(n) mod N for various values of N see A046694, A126811-...

Cf. A006352, A099059, A099060, A262339.

For primes p such that tau(p) == -1 mod 23 see A106867.

Cf. A010816, A004009. A013973.

Cf. A126832(n) = a(n) (mod 5).

Sequence in context: A211148 A265858 A282859 * A278577 A022716 A181104

Adjacent sequences:  A000591 A000592 A000593 * A000595 A000596 A000597




N. J. A. Sloane



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Last modified March 25 15:14 EDT 2017. Contains 284082 sequences.