Even Values of Ramanujan’s Tau-Function

Abstract

In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer α is a value of τ (n). For odd α, Murty, Murty, and Shorey proved that τ (n) = α for sufficiently large n. Several recent papers have identified explicit examples of odd α which are not tau-values. Here we apply these results (most notably the recent work of Bennett, Gherga, Patel, and Siksek) to offer the first examples of even integers that are not tau-values. Namely, for primes we find thatMoreover, we obtain such results for infinitely many powers of each prime 3 ≤ < 100. As an example, for = 97 we prove that τ (n) / ∈ {2 • 97 j : 1 ≤ j ≡ 0 (mod 44)} ∪ {−2 • 97 j : j ≥ 1}.

Some Remarks on Small Values of $τ(n)$

“…Shortly thereafter, Bennett, Gherga, Patel, and Siksek [5] proved that the same is true for any positive power of ℓ, and that τ (n) = ±3 a 5 b 7 c 11 d for any a, b, c, d ∈ Z ≥0 and any n > 1. Using some of these results concerning odd primes, Balakrishnan, Ono, and Tsai [4] were able to make progress eliminating even integers as values of the τ -function. For odd primes ℓ, they showed that…”

“…The proof of Theorem 1.1 follows the approach of [3] and [4] for odd and even values, respectively. For each α, we use the fact that smaller values have been excluded as values of the τ -function to show that τ (n) = α implies n = p a for some prime p. Due to the classification of primitive divisors of Lucas sequences in [6,1], we are able to restrict the list of possibilities for a, each of which we then eliminate using the congruences for the τ -function listed in [19].…”

“…This eliminates all remaining pairs (α, a) in Table 2 The main result of this section is the following theorem concerning even inadmissible values for the τ -function: for all n ∈ Z + . We generalize the proof of Theorem 1.1 in [4], and apply the following argument to the values α in (4.1) inductively starting with the α of smallest absolute value (for example, we need to know that 12 is not an admissible value for the τ -function in order to show that τ (n) = 24 for all n ∈ Z + ). Suppose for the sake of contradiction that for some α in (4.1), there exists a positive integer n such that τ (n) = α.…”