The Vanishing of Ramanujan's Function tau(n) - MathOverflow

The Vanishing of Ramanujan's Function tau(n)

2010-07-08T15:01:55Z

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.

Answer by Bruce Westbury for The Vanishing of Ramanujan's Function tau(n)

2010-07-08T15:31:53Z

I don't know if this helps but you can put $D=24$ in (13) (14) of my paper to get an explicit formula for $\tau(n)$.

MR2218820 (2007c:17009) Westbury, Bruce W. Universal characters from the Macdonald identities. Adv. Math. 202 (2006), no. 1, 50--63.

doi:10.1016/j.aim.2005.03.013

Since $SL(5)$ is a simple Lie algebra of dimension 24 this also relates $\tau(n)$ to the affine root system of type $A_4$.

I doubt Lehmer would have had this in mind.

Addendum I started this project with the following problem. Let $\mathfrak{g}$ be a simple Lie algebra whose dimension is $D$. Normalise the Casimir so it acts as 1 on $\mathfrak{g}$. Now consider the subspace of the exterior power $\wedge^k \mathfrak{g}$ on which the Casimir acts by $k$. This is a representation of $\mathfrak{g}$ but obviously does not make sense for $k>D$. Taking $\mathfrak{g}=\mathfrak{sl}(5)$ we have $\tau(k)$ is the dimension of a representation for small $k$ (certainly no more than 24). I doubt this is interesting.

The conclusion of the project was that for all $k$ there is a complex of representations of $\mathfrak{g}$. Then the Euler characteristic is a virtual representation. This can be written as a sum (with signs) of representations of $\mathfrak{g}$ using the MacDonald identities for affine $\mathfrak{g}$. This gives $\tau(k)$ as the dimension of a virtual representation of $\mathfrak{sl}(5)$ for all $k$.

Because of the signs this does not give an immediate solution to Lehmer's question. However it is a different way of looking at the problem.

I also give an explicit formula for $\tau(k)$ in terms of partitions and hooklengths. I believe this is new.

Answer by SandeepJ for The Vanishing of Ramanujan's Function tau(n)

2010-07-08T16:18:11Z

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

Answer by Charles Matthews for The Vanishing of Ramanujan's Function tau(n)

2010-07-08T18:24:13Z

A simple reason: this is a function of $n$ satisfying significant congruences. If it vanishes, that is further congruence information.