8

SCOTT AHLGREN AND KEN ONO

2. If s

2

1 and M

=

25

,

then for every 0

:S

i 25 we have

d(2

5

)

= ]__

t

2s

3. If s

2

1 and M

=

3

5

,

then for every 0

:S

i 35 we have

4. If there is a prime

£

2

5 for which

£

I

M, then for every 0

:S

r M we have

Virtually nothing is known about Conjecture 5.6. Part (1) is not known for

any values of rand M. If M is coprime to 6, then Theorem 2 implies that

liminf J

0

(M, X) 0.

X--oo

There are no other pairs of integers 0 r M for which it is known that

liminf

Or(M,

X) 0.

X--oo

When M

=

2, part (2) is the well known "folklore conjecture" studied by Parkin

and Shanks in the 1960s

[P-S].

In this direction, the best results are due to Serre

[N-R-S]

and the first author

[Al].

It is now known that

#{n

:S

X p(n) =: 0 (mod 2)}

»

Vx

#{n

:S

X p(n)

=

1 (mod 2)}

»

Vx/logX.

Obviously, this falls far short of Conjecture 5.6. The table below provides data

supporting parts (2) and (3) of Conjecture 5.6 when s

=

1.

X Jo(2; X) J1(2; X) Jo(3; X) Jl(3;X) J2(3; X)

200,000 0.5012 0.4988 0.3332 0.3331 0.3337

400,000 0.5000 0.5000 0.3339 0.3324 0.3336

600,000 0.5000 0.5000 0.3337 0.3326 0.3337

800,000 0.5006 0.4994 0.3331 0.3333 0.3336

1,000,000 0.5004 0.4996 0.3330 0.3336 0.3334

Although we have insufficient data to conjecture a value for

dr(M)

for any

0

:S

r

M with M coprime to 6, it is natural to consider the following problem.

PROBLEM

5. 7. Find a lower bound for do ( M) when M is coprime to 6.

Suppose that £

2

5 is prime. In view of Theorem 1, Theorem 2, and Con-

jecture 5.4, it is natural to consider the distribution of p(n) (mod£) for those n

(mod £)which do not belong to Se. Based on preliminary calculations, the following

speculation does not appear to be too far-fetched.