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. 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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