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.