On some claims in Ramanujan's `unpublished' manuscript on the partition and tau functions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of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently.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 $\tau (n)$ modulo some special primes q. He proves for example that $\tau (n)\equiv {\sum }_{d|n}{d}^{11}(\mathrm{mod}691)$. He defines $t_n=1$ if $\tau (n)$ is not divisible by q and $t_n=0$ otherwise. He then typically writes: "It can be shown by transcendental methods that $$\sum _{k=1}^n{t}_k=C{\int }_1^n\frac{dx}{(\log x{)}^{\delta }}+O(\frac{n}{(\log n{)}^r}).$$ where r is any positive number" (after stating some weaker estimates for the above sum). The number $\delta$ 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 $r>1+\delta$ 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.

14 pages, 1 table. Slightly revised version that will appear in The Ramanujan Journal