For any ε > 0 and sufficiently large prime q, in every residue class a coprime to q the Liouville function λ changes sign within the range up to q^{5/2 + ε}.
Xylouris,Linniks Konstante ist kleiner als 5, Chebyshevski˘ ı Sb.19, No
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Sign changes of the Liouville function in arithmetic progressions
For any ε > 0 and sufficiently large prime q, in every residue class a coprime to q the Liouville function λ changes sign within the range up to q^{5/2 + ε}.