2002/01/31 by Pieter Moree
Mathematics · #math.NT #msc:11N25
published as Ramanujan J. 8 (2004), 317-330. · 14 pages, 1 table. Slightly revised version that will appear in The Ramanujan Journal
arxiv created 2002/12/03 · arxiv updated 2009/11/30
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of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently.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)≡ ∑d|nd11(\rm mod691). He defines tn=1 if τ(n) is not divisible by q and tn=0 otherwise. He then typically writes: "It can be shown by transcendental methods that ∑k=1n tk=C∫1n 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.