Suppose , with completely multiplicative.
Then . Now if each of the coefficients is non-negative, this implies for example that each is non-negative. But each is a product of terms of the form , hence is also non-negative.
Let , for two real primitive characters . Multiplication of Euler products of these functions gives us
with . Davenport (in p.129 of Mult. Numb. Theory) claims that taking logarithms of Euler products gives
which is fine, but then claims that since the coefficients here are nonnegative, we have for all . Can someone show me the reasoning for this last fact? It doesn't seem obvious to me.
so the coefficients are actually not completely multiplicative.
Let G(s) = log F(s). Then F(s) = exp(G(s)). Express exp(G(s)) as a power series in G(s). Since the coefficients of the power series for the exponential function are positive, and the coefficients of G(s) are non-negative, then the coefficients of F(s) are non-negative. Does that work?