OFFSET
1,2
COMMENTS
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
REFERENCES
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.
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, Birkhäuser, 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. 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.
Van der Blij, F. "The function tau (n) of S. Ramanujan (an expository lecture)." Math. Student 18 (1950): 83-99.
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.
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.
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.
Bruce C. Berndt and Pieter Moree, Sums of two squares and the tau-function: Ramanujan's trail, arXiv:2409.03428 [math.NT], 2024.
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 [Broken link]
D. X. Charles, Computing The Ramanujan Tau Function
B. Cloitre, On the fractal behavior of primes, 2011. [Broken link]
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], (18-March-2014).
B. Edixhoven et al., Computing the coefficients of a modular form, arXiv:math/0605244 [math.NT], 2006-2010.
J. A. Ewell, Ramanujan's Tau Function, Proc. Amer. Math. Soc. 128 (2000), 723-726.
J. 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]
L. 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, arXiv:math/0607169 [math.NT], 2006.
Frank Garvan and Michael J. Schlosser, Combinatorial interpretations of Ramanujan’s tau function, arXiv:1606.08037 [math.CO], 2016; Discrete Mathematics 341.10 (2018): 2831-2840.
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.
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
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.
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.
N. Lygeros and O. Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Yuri Matiyasevich, Computational rediscovery of Ramanujan's tau numbers, Integers (2018) 18A, Article #A14.
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 and 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, 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
J. Perry, Ramanujan's Tau Function (broken link?)
Simon Plouffe, The first 225035 terms (432 MB)
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.
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, Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 9, no 1 (1967-1968), exp. no 14, pp. 1-17.
J.-P. 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.
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.
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
K. S. Williams, Historical remark on Ramanujan's tau function, Amer. Math. Monthly, 122 (2015), 30-35.
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
EXAMPLE
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
MAPLE
M := 50; t1 := series(x*mul((1-x^k)^24, k=1..M), x, M); A000594 := n-> coeff(t1, x, n);
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 *)
PROG
(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);
(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)={
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
(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
(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
(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
CROSSREFS
KEYWORD
sign,easy,core,mult,nice
AUTHOR
STATUS
approved