Pólya’s conjecture

(George Pólya) Given any range of consecutive integers from 1 to $n>1$, at least half, if not more, of the integers in that range will have an odd number of prime factors (not necessarily distinct). Or, restated using the Liouville function $\lambda (i)={(-1)}^{\mathrm{\Omega }(i)}$ (where $\mathrm{\Omega }(i)$ is the number of (nondistinct) prime factors function (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction)), there is no such $n>1$ such that $L(n)>0$ where

$$L(n)=\sum _{i=1}^n\lambda (i).$$

The zeroes of the sum of the Liouville function below 1000 (namely 2, 4, 6, 10, 16, 26, 40, 96, 586, listed in A028488 of Sloane’s OEIS) were known early on after the conjecture was posed. These are all followed by primes, with the exception of 26, which is followed by $3^3$.

Arthur Ingham proved the conjecture false in 1942 and gave a method for finding counterexamples, but the first counterexample wasn’t found until 1960 by Robert Lehman, namely $n=906180359$. Two decades later, Minoru Tanaka found the smallest counterexample at $n=906150257$; the next three integers have 2, 4 or 6 non-distinct prime factors.

Title	Pólya’s conjecture
Canonical name	PolyasConjecture
Date of creation	2013-03-22 16:39:43
Last modified on	2013-03-22 16:39:43
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Conjecture
Classification	msc 11A25
Synonym	Polya’s conjecture
Synonym	Polya conjecture
Synonym	Pólya conjecture