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.
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