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Pólya Conjecture

Let n be a positive integer and r(n) the number of (not necessarily distinct) prime factors of n (with r(1)=0). Let O(m) be the number of positive integers <=m with an odd number of prime factors, and E(m) the number of positive integers <=m with an even number of prime factors. Pólya (1919) conjectured that

L(m)=E(m)-O(m)
(1)
=sum_(n=1)^(m)lambda(n)
(2)

is <=0, where lambda(n) 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, L(906180359)=1, and the smallest counterexample m=906150257 was found by Tanaka (1980). It begins a run of counterexamples through 906150293 (OEIS A189229).

The first n for which L(n)=0 are n=2, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, 906150294, 906150308, 906150310, 906150314, 906151516, ... (Tanaka 1980, OEIS A028488). The accompanying OEIS b-file extends this list through a(317312)=352748014189958, and in particular confirms a(253)=351100332278250. It is unknown if L(x) changes sign infinitely often (Tanaka 1980).

This Pólya conjecture is unrelated to the Hilbert-Pólya conjecture, which concerns a spectral interpretation of the zeros of the Riemann zeta function.

See also

Andrica's Conjecture, Hilbert-Pólya Conjecture, Liouville Function, Prime Factor

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References

Brent, R. P. and van de Lune, J. "A Note on Pólya's Observation Concerning Liouville's Function." Dec. 20, 2011. https://arxiv.org/abs/1112.4911.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. 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.Veritasium and Fern. "The Obviously True Theorem No One Can Prove." 2025. https://www.youtube.com/watch?v=x32Zq-XvID4.

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Pólya Conjecture

Cite this as:

Weisstein, Eric W. "Pólya Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolyaConjecture.html

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