Parity of number of partitions of an integer

[Richard Guy]

`Ono' AND `partition' got 21 hits on Math Sci Net.
Here's the review MR 96d:11108.    R.

Subbarao conjectured that every arithmetic progression (AP) contains (a) 
infinitely many $n$ for which $p(n)$ is even and (b) infinitely many $n$ 
for which $p(n)$ is odd. In this fascinating paper, the author goes a long 
way towards settling this conjecture, showing that (a) is true for every 
AP and that (b) is true for every AP that contains at least one $n$ for 
which $p(n)$ is odd. He also gives a bound on how far we have to look for 
odd $p(n)$ to settle (b) and states that he has shown (by computer search) 
that (b) holds for all APs of numbers $\equiv r\bmod t$, for $0 \leq r < t 
\leq 10^5$.

The author uses the theory of modular forms, in particular, a well-known 
result (due to Serre) that almost all the Fourier coefficients of a 
modular form (of positive integer weight) are divisible by any given 
integer and a more recent result (due to Sturm), from which he finds the 
bound mentioned above. Using Hecke operators, he establishes (Lemma 1) an 
interesting fact, namely that a function $f(z) \equiv \sum_{1\leq i\leq 
s}\sum_{n=0}^{\infty}a_i(n)q^{wn^2 + b_i}\bmod m$, where $a_i(n)$ is not 
divisible by $m$ for infinitely many $n$, cannot be a modular form.

Reviewed by Richard P. Lewis


On Thu, 4 Sep 2003, benoit wrote:

> 
> Seems the previous link doesn't work. The paper was in :
> 
> ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY
> Volume 1, Issue 1, 1995
> 
> PARITY OF THE PARTITION FUNCTION
> by KEN ONO
> (Communicated by Don Zagier)
> 
> Benoit Cloitre