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.
Okay, this would work, but it presupposes that can be written in the form where the a_p are the same as in my original post. This product only arises from the Dirichlet series if the a_n are completely multiplicative. If we knew explicitly what the a_n's were we might be able to prove this. However I think it doesn't work, as using your notation we would need to put
so
but then
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?