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
 Verified by
 Software
 End date
 Duration
(11, 317)
 1810
 P
 Olivier Rozier
 PARI/GP
(17, 433)
 2924
 P
 Olivier Rozier
 PARI/GP
 12/02/2011
 10 hrs
(29, 31)
 242
 P
(29, 83)
 660
 P
(29, 229)
 1834
 P
 Olivier Rozier
 PARI/GP
(41, 2297)
 20367
 P
 François Morain
 ECPP
 06/04/2018
 24 days
(41, 28289) 250924 PRP
(47, 5)
 37
 P
(47, 47)
 424
 P
(47, 4177)
 38404
 P
 Andreas Enge
 CM/ECPP
 22/06/2022
 15 days
(59, 1381)
 13441
 P
 Gerasimos Politis**
 Primo
 12/10/2013
 83 days
(59, 8971)
 87365
 PRP
(79, 1571)
 16386
 P
 Gerasimos Politis**
 Primo
 07/02/2014
(79, 6317)
 65920
 PRP
(89, 73)
 772
 P
(97, 331)
 3606
 P
 Olivier Rozier
 PARI/GP
 16/02/2011
 31 hrs
(97, 887)
 9682
 P
 Keira Stambolidou**
 Primo
 26/05/2013
(*) 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 :
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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Ramanujan's tau function © 2010 Olivier Rozier