The conjecture was 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).
It begins a run of counterexamples through (OEIS A189229).
The first
for which
are ,
4, 6, 10, 16, 26, 40, 96, 586, 906150256, 906150294, 906150308, 906150310, 906150314,
906151516, ... (Tanaka 1980, OEIS A028488).
The accompanying OEIS b-file records 317312 known zeros through , and lists . These are known zeros, and the table
does not establish completeness beyond the first 252 terms (Borwein et al. 2008).
The function
changes sign infinitely often (Haselgrove 1958, Borwein et al. 2008).
Borwein, P.; Ferguson, R.; and Mossinghoff, M. J. "Sign Changes in Sums of the Liouville Function." Math. Comput.77,
1681-1694, 2008. https://doi.org/10.1090/S0025-5718-08-02036-X.Brent,
R. P. and van de Lune, J. "A Note on Pólya's Observation Concerning
Liouville's Function." 21 Dec 2011. https://arxiv.org/abs/1112.4911.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. Sequences A028488 and A189229 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. https://doi.org/10.3836/tjm/1270216093.Veritasium
and Fern. "The Obviously True Theorem No One Can Prove." 2025. https://www.youtube.com/watch?v=x32Zq-XvID4.
. 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. Their difference is
is the Liouville function.
, and the smallest counterexample
was found by Tanaka (1980).
It begins a run of counterexamples through 
(OEIS A189229).
for which 
are 
,
4, 6, 10, 16, 26, 40, 96, 586, 906150256, 906150294, 906150308, 906150310, 906150314,
906151516, ... (Tanaka 1980, OEIS A028488).
The accompanying OEIS b-file records 317312 known zeros through 
, and lists 
. These are known zeros, and the table
does not establish completeness beyond the first 252 terms (Borwein et al. 2008).
The function 
changes sign infinitely often (Haselgrove 1958, Borwein et al. 2008).
