Pólya conjecture

Let \(\lambda(n)\) be the, which is +1 if \(n\) has an even number of prime factors and -1 if \(n\) has an odd number of prime factors. The Pólya conjecture states that for all \(n > 1\):

\[L(n) = \sum_{i = 1}^{n} \lambda(i) \leq 0.\]

This can be explained visually as follows: Cedric and Royce are standing side by side. At time \(n\), Cedric takes a step forward if \(n\) has an even number of prime factors, and Royce steps forward if \(n\) has an odd number of prime factors. The Pólya conjecture is equivalent to the statement that Royce will always be ahead of Cedric after the starting time.

C. B. Haselgrove disproved the conjecture in 1958 by showing that the sum becomes positive. Tanaka found the smallest counterexample in 1980, \(L(906180359) = 1\). Defining a "crossover" as a point when \(L(n) = 1\) and \(L(n - 1) = 0\), larger crossovers have been found. It is unknown whether there are infinitely many of them.