On some claims in Ramanujan's `unpublished' manuscript on the partition and tau functions

Abstract

Towards the end of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently. Nevertheless, this manuscript gave rise to a lot of subsequent work. In it Ramanujan considers congruences for $τ(n)$ modulo some special primes q. He proves for example that $τ(n)\equiv \sum_{d|n}d^{11}({\rm mod}691)$. He defines $t_n=1$ if $τ(n)$ is not divisible by q and $t_n=0$ otherwise. He then typically writes: "It can be shown by transcendental methods that $$ \sum_{k=1}^n t_k=C\int_1^n {dx\over (\log x)^δ}+O({n\over (\log n)^r}). $$ where r is any positive number" (after stating some weaker estimates for the above sum). The number $δ$ is a positve rational number depending on q and for the positive number C Ramanujan usually wrote down an Euler product. In this paper it is shown that Ramanujan's claim for every $r>1+δ$ and each of the special primes q is false. Furthermore, we correct a 1928 paper of Geraldine Stanley who claimed to have disproved Ramanujan's claim in case q=5.