 Ramanujan's tau function It looks simple, isn't it ?
But computation of τ(n) is not ...

First values are: (1) = 1 (2) = -24 (3) = 252 (4) = -1472 (5) = 4830 (6) = -6048

Ramanujan found it has remarkable properties: for m, n coprime integers, for p prime.

Now we need a formula to compute τ(p) for all primes p.

A formula related to Catalan triangle

From Eichler-Selberg trace formula, one may derive where p is prime, and H(n) is Hurwitz-Kronecker class number of binary quadratic forms of negative discriminant -n.

A new and faster formula

Recombining several traces leads to the formula where p is prime and H5(n)=n5H(n).

Note that it mainly consists of a finite sum in the table of H5(n) integers. Hence it is a somewhat faster algorithm when computing τ(p) for all primes.

After rearrangement, I obtained the nice formulation where C5 = 42 is the 5th Catalan number.

Non-ordinary primes

We found that the only primes p for which τ(p) ≡ 0 (mod p)  are 2, 3, 5, 7, 2411 and 7758337633, up to 1010.

See OEIS entry: A007659

Odd prime values

Our purpose is to identify the integers n for which τ(n) is an odd prime, disregarding the sign of τ(n). The main result is the following :

Theorem
Let n be a positive integer such that τ(n) is an odd prime. Then n is of the form pq-1 where p and q are odd primes and p is ordinary.

One sets LR(p, q) := τ(pq-1), where "LR" stands for Lehmer-Ramanujan.

Here we give all known pairs (p,q), p < 100, such that LR(p, q) is prime (P) or probable prime (PRP):

 (p, q) Digits* Primality (11, 317) 1810 P (17, 433) 2924 P (29, 31) 242 P (29, 83) 660 P (29, 229) 1834 P (41, 2297) 20367 PRP (41, 28289) 250924 PRP (47, 5) 37 P (47, 47) 424 P (47, 4177) 38404 PRP (59, 1381) 13441 P (59, 8971) 87365 PRP (79, 1571) 16386 P (79, 6317) 65920 PRP (89, 73) 772 P (97, 331) 3606 P (97, 887) 9682 P
(*) number of decimal digits of LR(p, q)

The prime values LR(59,1381), LR(79,1571) and LR(97,887) have been certified by G. Politis and K. Stambolidou, using Primo ECPP software. Light gray areas are currently being processed. Blue areas have been mostly processed by the LR team : Anna, Athina, Dimitris, Elias, Fotini, Giorgos, Keira, Kostas, Koulla, Marios, Neoklis, Patrice, Theodora, Vicky, Xrisa. Thanks to all of them !

LR_data.pdf : known (probable) prime values LR(p,q) with p<1000

Numerical data

Tau_0001000000.zip (1.4 Mb): tau(p) for all primes p < 10^6
Tau_0010000000.zip (13.4 Mb): tau(p) for all primes p < 10^7

Publications

N. Lygeros & O. Rozier, Odd prime values of the Ramanujan tau function, Ramanujan Journal, 10.1007/s11139-012-9420-8 (2013)
N. Lygeros & O. Rozier, A new solution to the equation τ(p) ≡ 0 (mod p), Journal of Integer Sequences 13, article 10.7.4 (2010)