OFFSET
1,1
COMMENTS
From Wolfdieter Lang, May 15 2016: (Start)
This sequence determines all values of Ramanujan's tau function A000594 due to alpha-multiplicativity with alpha(x) = x^11 (the weight of the modular cusp form eta^{24}(z) with the Dedekind eta function is k = 12). See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 114, eq. (3) for the tau function. This implies multiplicativity of tau with tau(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. See the Apostol Exercise 6 on p. 139.
Note that the product representation of the Dirichlet series Sum_{n >=1} tau(n)/Sum_{n >= 1} tau(n)/n^s = Prod_{n >= 1} 1/(1 - a(n)/prime(n)^s + prime(n)^(11) / prime(n)^(2*s)) (see the Mordell reference, eq. (2)) leads also to this formula for tau(p^k) for primes p after expanding the factors of the product and collecting powers of 1/p^(k*s). If one insists on convergence of the product one can use s >= 7, if one uses Ramanujan's 1916 conjecture (proved by P. Deligne 1974) |tau(p)| <= 2*p^(11/2), i.e., |a(n)| <= 2*sqrt(prime(n)^11).
(End)
REFERENCES
Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 114, 138-139.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Denis Xavier Charles, Computing the Ramanujan tau function, Ramanujan J. 11:2 (2006), pp. 221-224.
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]
Louis J. Mordell, On Mr. Ramanujan's empirical expansions of modular functions, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
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.
Wikipedia, Ramanujan-Petersson conjecture
FORMULA
a(n)*a(m) = A000594(prime(n)*prime(m)) for n != m (from the tau multiplicativity). - Wolfdieter Lang, May 15 2016
a(n)^2 = A000594(prime(n)^2)) + prime(n)^11 (from alpha-multiplicativity). - Wolfdieter Lang, May 15 2016
EXAMPLE
84480 = A000594(2^3) = sqrt(2^(11))^3*S(3, -24/sqrt(2^(11))) = (-24)*((-24)^2 -2*2^11) = 84480. - Wolfdieter Lang, May 15 2016
MATHEMATICA
RamanujanTau[Prime[Range[30]]] (* Jean-François Alcover, Dec 01 2015 *)
PROG
(PARI) taup(p)=(65*sigma(p, 11)+691*sigma(p, 5)-691*252*sum(k=1, p-1, sigma(k, 5)*sigma(p-k, 5)))/756
a(n)=taup(prime(n)) \\ Charles R Greathouse IV, Apr 22 2013
(PARI) H(n)=sumdiv(core(n, 1)[2], d, my(D=-n/d^2); if(D%4<2, qfbclassno(D)/max(1, D+6)))
taup(p)=my(x='x, P=x^5-9*p*x^4+28*p^2*x^3-35*p^3*x^2+15*p^4*x-p^5); p^5*H(4*p)/2-1-sum(t=1, sqrtint(4*p), subst(P, x, t^2)*H(4*p-t^2))
a(n)=taup(prime(n)) \\ Charles R Greathouse IV, Apr 25 2013
(Perl) use ntheory ":all"; forprimes { say ramanujan_tau($_) } 100 # Dana Jacobsen, Sep 05 2015
(Sage)
[p for (n, p) in enumerate(list(delta_qexp(100))) if is_prime(n)] # Peter Luschny, May 16 2016
(Python)
from sympy import prime, divisor_sigma
def A076847(n): return -24 if n == 1 else (q:=(p:=prime(n))**4)*(p+1)-24*(sum((i*(i*(i*(70*i - 140*p) + 90*p**2) - 20*p**3) + q)*divisor_sigma(i)*divisor_sigma(p-i) for i in range(1, p+1>>1))) # Chai Wah Wu, Nov 09 2022
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Nov 23 2002
STATUS
approved