Pólya conjecture

File:Liouville-big.svg

Summatory Liouville function L(n) up to n = 10⁷. The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function.

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Closeup of the summatory Liouville function L(n) in the region where the Pólya conjecture fails to hold.

File:Liouville-log.svg

Logarithmic graph of the negative of the summatory Liouville function L(n) up to n = 2 × 10⁹. The green spike shows the function itself (not its negative) in the narrow region where the conjecture fails; the blue curve shows the oscillatory contribution of the first Riemann zero.

In number theory, the Pólya conjecture stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an odd number of prime factors. The conjecture was posited by the Hungarian mathematician George Pólya in 1919, and proved false in 1958 by C. Brian Haselgrove. The size of the smallest counterexample is often used to show how a conjecture can be true for many numbers, and still be false.

Statement

Pólya's conjecture states that for any n (> 1), if we partition the natural numbers less than or equal to n (excluding 0) into those with an odd number of prime factors, and those with an even number of prime factors, then the former set has at least as many members as the latter set. (Repeated prime factors are counted the requisite number of times—thus 24 = 2³ × 3¹ has 3 + 1 = 4 factors i.e. an even number of factors, while 30 = 2 × 3 × 5 has 3 factors, i.e. an odd number of factors.)

Equivalently, it can be stated in terms of the summatory Liouville function, the conjecture being that

$L(n) = \sum_{k=1}^n \lambda(k) \leq 0$

for all n > 1. Here, λ(k) = (−1)^Ω(k) is positive if the number of prime factors of the integer k is even, and is negative if it is odd. The big Omega function counts the total number of prime factors of an integer.

Disproof

Pólya's conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10³⁶¹.

An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960; the smallest counterexample is n = 906,150,257, found by Minoru Tanaka in 1980.

The Pólya conjecture fails to hold for most values of n in the region of 906,150,257 ≤ n ≤ 906,488,079. In this region, the summatory Liouville function reaches a maximum value of 829 at n = 906,316,571.

References

• Weisstein, Eric W., "Pólya Conjecture", MathWorld.