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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)=sum_(n=1)^mlambda(n)

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). The first n for which L(n)=0 are n=2, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ... (Tanaka 1980, OEIS A028488). It is unknown if L(x) changes sign infinitely often (Tanaka 1980).


See also

Andrica's Conjecture, Liouville Function, Prime Factor

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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.

Cite this as:

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

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