Ramanujan's tau function

<Tau definition>

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

First values are:
Tau(1) = 1
Tau(2) = -24
Tau(3) = 252
Tau(4) = -1472
Tau(5) = 4830
Tau(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
<Tau formula>
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
<Tau fast 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
<Tau faster formula>
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)


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Math page © 2010 Olivier Rozier