This paper is concerned with the constancy in the sign of L(X,α)=∑1Xnαλ(n), where λ(n) the Liouville
function. The non-positivity of L(X,0) is the P\'{o}lya conjecture, and the
non-negativity of L(X,1) is the Tur\'{a}n conjecture --- both of which are
false. By constructing an auxiliary function, evidence is provided that L(X,21) is the best contender for constancy in sign. The core of this
paper is the conjecture that L(X,21)≤0 for all X≥17:
this has been verified for X≤300,001.Comment: 5 page