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.
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 :
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):
(*) number of decimal digits of LR(p, q)
(**) supervised by Nik Lygeros and Olivier Rozier
and the distribution of such values, for p < 1000, on a semi-log scale :
LR_data.pdf : known (probable) prime values LR(p,q) with p<1000
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
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)
Ramanujan's tau function © 2010 Olivier Rozier