A076847 Ramanujan function tau(p) as p runs through the primes.

-24, 252, 4830, -16744, 534612, -577738, -6905934, 10661420, 18643272, 128406630, -52843168, -182213314, 308120442, -17125708, 2687348496, -1596055698, -5189203740, 6956478662, -15481826884, 9791485272, 1463791322, 38116845680, -29335099668, -24992917110

Offset: 1

Author

N. J. A. Sloane, Nov 23 2002

Keywords

• sign

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)

Examples

			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
		

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

Crossrefs

Cf. A000594, A049310, A278577 (prime powers).

Programs

Mathematica

RamanujanTau[Prime[Range[30]]] (* Jean-François Alcover, Dec 01 2015 *)

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

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

Sage

[p for (n,p) in enumerate(list(delta_qexp(100))) if is_prime(n)] # Peter Luschny, May 16 2016

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