The key to your question is [lacunarity][1] in modular functions.

The tau function, as we know, occurs as the coefficient of the [Discriminant function][2], which in turn is the 24th power of the [Eta function][3]. 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


  [1]: https://en.wikipedia.org/wiki/Lacunary_function
  [2]: https://planetmath.org/ModularDiscriminant
  [3]: https://mathworld.wolfram.com/DedekindEtaFunction.html