A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).
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
Offset: 1
Examples
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
References
- Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 114, 131.
- Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
- Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, AMS 2001; see p. 298.
- Nathan 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.
- 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.
- Yu. I. Manin, Mathematics and Physics, Birkhäuser, Boston, 1981.
- Henry McKean and Victor Moll, Elliptic Curves, Camb. Univ. Press, 1999, 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.
- Srinivasa Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 153, Ed. G. H. Hardy et al., AMS Chelsea 2000.
- Srinivasa Ramanujan, On Certain Arithmetical Functions. Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
- Jean-Pierre Serre, A course in Arithmetic, Springer-Verlag, 1973, see p. 98.
- Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, 1994, 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.
- Don Zagier, Introduction to Modular Forms, Chapter 4 in M. Waldschmidt et al., editors, From Number Theory to Physics, Springer-Verlag, 1992.
- Don Zagier, "Elliptic modular forms and their applications", in: The 1-2-3 of modular forms, Springer Berlin Heidelberg, 2008, pp. 1-103.
Links
- Simon Plouffe, Table of n, a(n) for n = 1..16090
- Jennifer S. Balakrishnan, William Craig, and Ken Ono, Variations of Lehmer's Conjecture for Ramanujan's tau-function, arXiv:2005.10345 [math.NT], 2020.
- Jennifer S. Balakrishnan, Ken Ono, and Wei-Lun Tsai, Even values of Ramanujan's tau-function, arXiv:2102.00111 [math.NT], 2021.
- Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary.
- Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
- Bruce C. Berndt and Pieter Moree, Sums of two squares and the tau-function: Ramanujan's trail, arXiv:2409.03428 [math.NT], 2024.
- Matthew Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory, Vol. 98, No. 2 (2003), 377-389. MR1955423 (2003k:11071).
- François Brunault, La fonction Tau de Ramanujan. [Wayback Machine link]
- Denis Xavier Charles, Computing The Ramanujan Tau Function.
- Benoit Cloitre, On the fractal behavior of primes, 2011.
- John Cremona, Home page.
- Maarten Derickx, Mark van Hoeij, and Jinxiang Zeng, Computing Galois representations and equations for modular curves X_H(l), arXiv:1312.6819 [math.NT], 2013-2014.
- Bas Edixhoven, Jean-Marc Couveignes, Robin de Jong, Franz Merkl, and Johan Bosman, Computing the coefficients of a modular form, arXiv:math/0605244 [math.NT], 2006-2010.
- John A. Ewell, Ramanujan's Tau Function, Proc. Amer. Math. Soc. 128 (2000), 723-726.
- John A. Ewell, Ramanujan's Tau Function.
- Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
- Luis H. Gallardo, On some formulae for Ramanujan's tau function, Rev. Colomb. Matem. 44 (2010) 103-112.
- M. Z. Garaev, V. C. Garcia, and S. V. Konyagin, Waring problem with the Ramanujan tau function, Izvestiya: Mathematics, Vol. 72, No 1 (2008), pp. 35-46; arXiv preprint, arXiv:math/0607169 [math.NT], 2006.
- Frank Garvan and Michael J. Schlosser, Combinatorial interpretations of Ramanujan's tau function, Discrete Mathematics, Vol. 341, No. 10 (2018), pp. 2831-2840; arXiv preprint, arXiv:1606.08037 [math.CO], 2016.
- Hansraj Gupta, The Vanishing of Ramanujan's Function(n), Current Science, 17 (1948), p. 180. [Wayback Machine link]
- James Lee Hafner and Jeffrey Stopple, A Heat Kernel Associated to Ramanujan's Tau Function, The Ramanujan Journal, Vol. 4, No. 2 (2000), pp. 123-128.
- Yang-Hui He and John McKay, Moonshine and the Meaning of Life, in: M. Bhargava et al., Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemporary Mathematics, Vol. 694, American Mathematical Society, 2017; arXiv preprint, arXiv:1408.2083 [math.NT], 2014.
- Michael J. Hopkins, Algebraic topology and modular forms, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317; arXiv preprint, arXiv:math/0212397 [math.AT], 2002.
- Masanobu Kaneko and Don 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
- Jon Keating and Brady Haran, The Key to the Riemann Hypothesis, Numberphile video (2016).
- Jerry B. Keiper, Ramanujan's Tau-Dirichlet Series [Dead link?]
- Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
- N. Laptyeva and V. K. Murty, Fourier coefficients of forms of CM-type, Indian Journal of Pure and Applied Mathematics, Volume 45, Issue 5 (October 2014), pp. 747-758.
- 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 orbit 1.12.a.a
- Florian Luca and Igor E. Shparlinski, Arithmetic properties of the Ramanujan function, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, Vol. 116. No. 1 (2006), pp. 1-8; arXiv preprint, arXiv:math/0607591 [math.NT], 2006.
- Nik Lygeros and Olivier Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010), Article 10.7.4.
- Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc., Vol. 348, No. 12 (1996), 4825-4856, see page 4852 Table I.
- Yuri Matiyasevich, Computational rediscovery of Ramanujan's tau numbers, Integers (2018) 18A, Article #A14.
- Keith Matthews, Computing Ramanujan's tau function.
- Stephen 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.
- Stephen 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.
- Pieter Moree, On some claims in Ramanujan's 'unpublished' manuscript on the partition and tau functions, arXiv:math/0201265 [math.NT], 2002.
- M. Ram Murty and V. Kumar Murty, The Ramanujan tau-function, in: The mathematical legacy of Srinivasa Ramanujan (Springer, 2012), p 11-23.
- M. Ram Murty, V. Kumar Murty, and T. N. Shorey, Odd values of the Ramanujan tau-function, Bulletin de la S. M. F., tome 115 (1987), p. 391-395.
- 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. [broken link]
- Jon Perry, Ramanujan's Tau Function. [Wayback Machine link]
- Simon Plouffe, The first 225035 terms (432 MB) [Wayback Machine link]
- Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.
- Srinivasa Ramanujan, Collected Papers, Table of tau(n);n=1 to 30.
- Jean-Pierre Serre, An interpretation of some congruences concerning Ramanujan's tau function, Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 9, no 1 (1967-1968), exp. no 14, pp. 1-17.
- Jean-Pierre Serre, An interpretation of some congruences concerning Ramanujan's Tau function, 1997.
- Jean-Pierre Serre, Sur la lacunarité des puissances de eta, Glasgow Math. Journal, 27 (1985), 203-221.
- 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, 2016.
- David A. Steffen, Les Coefficients de Fourier de la forme modulaire: La fonction de Ramanujan tau(n), 1998.
- William A. Stein, Database.
- Zhi-Wei Sun, Questions on the reciprocals of the values of the tau function, Question 484763 at MathOverflow, December 25, 2024.
- Zhi-Wei Sun, Pythagorean triples and Ramanujan's tau function, Question 484936 at MathOverflow, December 28, 2024.
- Zhi-Wei Sun, Additive combinatorics for Ramanujan's tau function, Question 485138 at MathOverflow, January 1, 2025.
- 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.
- F. Van der Blij, The function tau (n) of S. Ramanujan (an expository lecture), Math. Student, Vol. 18, No. 3 (1950), pp. 83-99; entire issues 3 and 4.
- Jan Vonk, Overconvergent modular forms and their explicit arithmetic, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.
- 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.
- Kenneth S. Williams, Historical remark on Ramanujan's tau function, Amer. Math. Monthly, 122 (2015), 30-35; author's copy.
- 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.
Crossrefs
Programs
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Julia
using Nemo function DedekindEta(len, r) R, z = PolynomialRing(ZZ, "z") e = eta_qexp(r, len, z) [coeff(e, j) for j in 0:len - 1] end RamanujanTauList(len) = DedekindEta(len, 24) RamanujanTauList(28) |> println # Peter Luschny, Mar 09 2018 -
Magma
M12:=ModularForms(Gamma0(1),12); t1:=Basis(M12)[2]; PowerSeries(t1[1],100); Coefficients($1);
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Magma
Basis( CuspForms( Gamma1(1), 12), 100)[1]; /* Michael Somos, May 27 2014 */
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Maple
M := 50; t1 := series(x*mul((1-x^k)^24,k=1..M),x,M); A000594 := n-> coeff(t1,x,n);
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Mathematica
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 *) -
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)={ if(e==1, (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
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PARI
a(n)=ramanujantau(n) \\ Charles R Greathouse IV, May 27 2016
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Python
from sympy import divisor_sigma def A000594(n): return n**4*divisor_sigma(n)-24*((m:=n+1>>1)**2*(0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + n**4)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) # Chai Wah Wu, Nov 08 2022
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Ruby
def s(n) s = 0 (1..n).each{|i| s += i if n % i == 0} s end def A000594(n) ary = [1] a = [0] + (1..n - 1).map{|i| s(i)} (1..n - 1).each{|i| ary << (1..i).inject(0){|s, j| s - 24 * a[j] * ary[-j]} / i} ary end p A000594(100) # Seiichi Manyama, Mar 26 2017 -
Ruby
def A000594(n) ary = [0, 1] (2..n).each{|i| s, t, u = 0, 1, 0 (1..n).each{|j| t += 9 * j u += j break if i <= u s += (-1) ** (j % 2 + 1) * (2 * j + 1) * (i - t) * ary[-u] } ary << s / (i - 1) } ary[1..-1] end p A000594(100) # Seiichi Manyama, Nov 25 2017
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Sage
CuspForms( Gamma1(1), 12, prec=100).0; # Michael Somos, May 28 2013
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Sage
list(delta_qexp(100))[1:] # faster Peter Luschny, May 16 2016
Formula
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(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 = Product_{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).
a(1) = 1, a(n) = -(24/(n-1))*Sum_{k=1..n-1} A000203(k)*a(n-k) for n > 1. - Seiichi Manyama, Mar 26 2017
G.f.: x*exp(-24*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Euler Transform of [-24, -24, -24, -24, ...]. - Simon Plouffe, Jun 21 2018
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). [See Douglas Niebur link]. - Wesley Ivan Hurt, Jul 22 2025
Comments