Pólya Conjecture
Let 
be a positive
integer and 
the number of (not necessarily distinct)
prime factors of 
(with 
). Let 
be the number of positive
integers 
with an odd
number of prime factors, and 
the number
of positive integers 
with an even number of prime factors.
Pólya (1919) conjectured that
 |
is 
, where 
is the
Liouville function.
The conjecture was made in 1919, and disproven by Haselgrove (1958) using a method due to Ingham (1942). Lehman (1960) found the first explicit counterexample, 
, and the smallest counterexample 
was found by Tanaka (1980). The first
for which 
are 
, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ...
(Tanaka 1980, OEIS A028488). It is unknown
if 
changes sign infinitely often (Tanaka
1980).
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