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).
Haselgrove, C. B. "A Disproof of a Conjecture of Pólya." Mathematika5, 141-145, 1958.Ingham,
A. E. "On Two Conjectures in the Theory of Numbers." Amer. J. Math.64,
313-319, 1942.Lehman, R. S. "On Liouville's Function."
Math. Comput.14, 311-320, 1960.Pólya, G. "Verschiedene
Bemerkungen zur Zahlentheorie." Jahresber. deutschen Math.-Verein.28,
31-40, 1919.Sloane, N. J. A. Sequence A028488
in "The On-Line Encyclopedia of Integer Sequences."Tanaka,
M. "A Numerical Investigation on Cumulative Sum of the Liouville Function"
[sic]. Tokyo J. Math.3, 187-189, 1980.
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
,
where 
is the Liouville function.
, 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).
