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 numberMathworldPlanetmathPlanetmath of prime factorsMathworldPlanetmath (not necessarily distinct). Or, restated using the Liouville functionDlmfMathworldPlanetmath λ(i)=(-1)Ω(i) (where Ω(i) is the number of (nondistinct) prime factors functionMathworldPlanetmath (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction)), there is no such n>1 such that L(n)>0 where


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 33.

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 conjectureMathworldPlanetmath
Synonym Pólya conjecture