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A000594 Ramanujan's tau function (or tau numbers).
(Formerly M5153 N2237)
105
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)
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.

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.

REFERENCES

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

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

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

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

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.

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

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

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

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

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

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.

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 lacunarit\'e 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, 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.

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.

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

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

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

LINKS

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...

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

B. Edixhoven et al., Computing the coefficients of a modular form

J. A. Ewell, Ramanujan's Tau Function

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

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

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

F. Luca and I. E. Shparlinski, Arithmetic properties of the Ramanujan function

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.

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

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

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

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

William Stein, Database

Eric Weisstein's World of Mathematics, Tau Function

Index entries for "core" sequences

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

FORMULA

G.f.: x * prod(k >= 1, (1 - x^k)^24 ).

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 formulae, 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

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 + ...

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 (dean.hickerson(AT)yahoo.com), 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-Fran├žois 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 *)

PROG

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

(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

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

CROSSREFS

Cf. A076847 (tau(p)), A037955, A027364, A037945, A037946, A037947, A008408 (Leech).

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

Cf. A006352.

Sequence in context: A052732 A086603 A211148 * A022716 A181104 A051828

Adjacent sequences:  A000591 A000592 A000593 * A000595 A000596 A000597

KEYWORD

sign,easy,core,mult,nice,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 20 18:30 EDT 2014. Contains 240820 sequences.