This is a problem I had a look at some years ago but always had the feeling that I was missing something behind its motivation.
D.H. Lehmer says in his 1947 paper, “The Vanishing of Ramanujan's Function τ(n),” that it is natural to ask whether τ(n)=0 for any n>0.
My question is: Why is it natural to wonder whether τ(n)=0 any n>0?
Are there any particular arithmetic properties among the many satisfied by τ(n) that would lead one to ponder its vanishing? The problem is mentioned here, where it's stated that it was a conjecture of Lehmer, although it's not actually presented as a conjecture in his paper, more a curiosity.
Maybe there is no deep reason to ponder the vanishing of τ(n), in which case that would be a satisfactory answer too.
Best Answer
The key to your question is lacunarity in modular functions.
The tau function, as we know, occurs as the coefficient of the Discriminant function, which in turn is the 24th power of the Eta function. The Eta function was known to be lacunary (having gaps or zero coefficients). Therefore it was natural for Lehmer in 1947 to wonder if coefficients of powers of eta are also zero. See the opening passage of the following paper
MR0021027 (9,12b) Lehmer, D. H. The vanishing of Ramanujan's function $\tau(n)$. Duke Math. J. 14, (1947). 429--433. http://projecteuclid.org/euclid.dmj/1077474140