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.