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 ofPrime factors, and the number of Positive Integers with an Even number ofPrime factors. Pólya conjectured that

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 wasfound by Tanaka (1980). The first for which are , 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ... (Tanaka 1980, Sloane's 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.

Sloane, N. J. A. Sequence A028488in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.

Tanaka, M. ``A Numerical Investigation on Cumulative Sum of the Liouville Function'' [sic]. Tokyo J. Math. 3, 187-189, 1980.

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