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).

REFERENCES:

Haselgrove, C. B. "A Disproof of a Conjecture of Pólya." Mathematika 5, 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.