Ramanujan's tau functionIt 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 H _{5}(n)=n^{5}H(n).Note that it mainly consists of a finite sum in the table of H _{5}(n) integers.
Hence it is a somewhat faster algorithm when computing τ(p) for all
primes.After rearrangement, I obtained the nice formulation where C _{5} = 42 is the 5^{th} 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 10 ^{10}.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 p ^{q-1} where p and q are odd primes and p is ordinary.One sets LR(p, q) := τ(p ^{q-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)
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
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