On a certain class of multiplicative functions

Let χ0, χ1, χ2, … be the sequence of all Dirichlet characters (in which the principal character χ0 occurs only once), ordered with increasig moduli. Define, for a sequence (αk)kεN of real numbers (αk→0 for j→∞, sufficiently quickly) the multiplicative function f by∑nf(n)ns=ζ−1(s)∏k=1∞(L(s,χk))akIt is the aim of the paper to construct sequences (αk) such that the following statements are equivalento(i)∑n⩽xf(n)=O(x1/2+ɛ)for every ɛ > 0,(ii)the Riemann hypothesis is true for all function L(s, χ)