CONGRUENCES AND CONJECTURES FOR THE PARTITION FUNCTION g
SPECULATION
5.8. If£
~
5 is prime and 0 :S; r
£,
then define
0~(£,
X) by
0, (£X)·= #{n X : p(n)
=
r (mod£) and n (mod£) rj. S£}
r ' ·
#{n X : n (mod£)
t/.
Se}
1
For every 0 :::; r £, is it true that lim
o~(£,
X) =
-?
X-oo
£
To support this speculation, we give data describing these functions when £ = 5.
X
156(5; X)
o~(5;X) o~(5;
X)
oH5;
x)
o~(5;X)
200,000 0.2006 0.1987 0.2011 0.2007 0.1988
400,000 0.2000 0.1996 0.2001 0.2007 0.1996
600,000 0.1999 0.1992 0.1995 0.2009 0.2006
800,000 0.1995 0.1996 0.1994 0.2010 0.2005
1,000,000 0.1999 0.1996 0.1997 0.2006 0.2002
References
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