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%I #41 Nov 09 2022 14:19:07
%S -24,252,4830,-16744,534612,-577738,-6905934,10661420,18643272,
%T 128406630,-52843168,-182213314,308120442,-17125708,2687348496,
%U -1596055698,-5189203740,6956478662,-15481826884,9791485272,1463791322,38116845680,-29335099668,-24992917110
%N Ramanujan function tau(p) as p runs through the primes.
%C From _Wolfdieter Lang_, May 15 2016: (Start)
%C 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.
%C 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).
%C (End)
%D Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 114, 138-139.
%H Charles R Greathouse IV, <a href="/A076847/b076847.txt">Table of n, a(n) for n = 1..10000</a>
%H Denis Xavier Charles, <a href="http://www.cs.wisc.edu/~cdx/CompTau.pdf">Computing the Ramanujan tau function</a>, Ramanujan J. 11:2 (2006), pp. 221-224.
%H D. H. Lehmer, <a href="http://dx.doi.org/10.1215/S0012-7094-47-01436-1">The Vanishing of Ramanujan's Function tau(n)</a>, Duke Mathematical Journal, 14 (1947), pp. 429-433.
%H D. H. Lehmer, <a href="/A000594/a000594.pdf">The Vanishing of Ramanujan's Function tau(n)</a>, Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]
%H Louis J. Mordell, <a href="http://www.archive.org/stream/proceedingsofcam1920191721camb#page/n133">On Mr. Ramanujan's empirical expansions of modular functions</a>, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
%H H. P. F. Swinnerton-Dyer, <a href="http://dx.doi.org/10.1007/978-3-540-37802-0_1">On l-adic representations and congruences for coefficients of modular forms</a>, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
%H Jan Vonk, <a href="https://doi.org/10.1090/bull/1700">Overconvergent modular forms and their explicit arithmetic</a>, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture">Ramanujan-Petersson conjecture</a>
%F a(n)*a(m) = A000594(prime(n)*prime(m)) for n != m (from the tau multiplicativity). - _Wolfdieter Lang_, May 15 2016
%F a(n)^2 = A000594(prime(n)^2)) + prime(n)^11 (from alpha-multiplicativity). - _Wolfdieter Lang_, May 15 2016
%e 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
%t RamanujanTau[Prime[Range[30]]] (* _Jean-François Alcover_, Dec 01 2015 *)
%o (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
%o a(n)=taup(prime(n)) \\ _Charles R Greathouse IV_, Apr 22 2013
%o (PARI) H(n)=sumdiv(core(n,1)[2],d,my(D=-n/d^2);if(D%4<2,qfbclassno(D)/max(1,D+6)))
%o 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))
%o a(n)=taup(prime(n)) \\ _Charles R Greathouse IV_, Apr 25 2013
%o (Perl) use ntheory ":all"; forprimes { say ramanujan_tau($_) } 100 # _Dana Jacobsen_, Sep 05 2015
%o (Sage)
%o [p for (n,p) in enumerate(list(delta_qexp(100))) if is_prime(n)] # _Peter Luschny_, May 16 2016
%o (Python)
%o from sympy import prime, divisor_sigma
%o 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
%Y Cf. A000594, A049310, A278577 (prime powers).
%K sign
%O 1,1
%A _N. J. A. Sloane_, Nov 23 2002
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