Pólya conjecture

The conjecture was set forth by the Hungarian mathematician George Pólya in 1919,[1] and proved false in 1958 by C. Brian Haselgrove.

The size of the smallest counterexample is often used to demonstrate the fact that a conjecture can be true for many cases and still fail to hold in general,[2] providing an illustration of the strong law of small numbers.

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.

In this region, the summatory Liouville function reaches a maximum value of 829 at n = 906,316,571.

Summatory Liouville function L ( n ) up to n = 10 7 . The (disproved) conjecture states that this function is always negative. The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function .
Closeup of the summatory Liouville function L ( n ) in the region where the Pólya conjecture fails to hold.
Logarithmic graph of the negative of the summatory Liouville function L ( n ) up to n = 2 × 10 9 . 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.