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Math Help - Coefficients of a Dirichlet series

  1. #1
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    Coefficients of a Dirichlet series

    Let F(s) = \zeta(s)L(s,\chi_1)L(s,\chi_2)L(s,\chi_1\chi_2), for two real primitive characters \chi_1, \chi_2. Multiplication of Euler products of these functions gives us

    F(s) = \displaystyle \sum_{n=1}^\infty a_n n^{-s}

    with a_1=1. Davenport (in p.129 of Mult. Numb. Theory) claims that taking logarithms of Euler products gives

    \displaystyle \log F(s) = \sum_p \sum_{m=1}^\infty \big(1+\chi_1(p^m)\big)\big(1+\chi_2(p^m)\big)m^{-1} p^{-ms},

    which is fine, but then claims that since the coefficients here are nonnegative, we have a_n \geqslant 0 for all n. Can someone show me the reasoning for this last fact? It doesn't seem obvious to me.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Suppose f(z)=\prod_p(1-g(p)p^{-s})^{-1}, with g completely multiplicative.

    Then \log f(z) = \sum_{p}\sum_{n=1}^\infty g(p^n)n^{-1}p^{-ns}. Now if each of the coefficients is non-negative, this implies for example that each g(p) is non-negative. But each g(n) is a product of terms of the form g(p), hence is also non-negative.
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  3. #3
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    Quote Originally Posted by Bruno J. View Post
    Suppose f(z)=\prod_p(1-g(p)p^{-s})^{-1}, with g completely multiplicative.

    Then \log f(z) = \sum_{p}\sum_{n=1}^\infty g(p^n)n^{-1}p^{-ns}. Now if each of the coefficients is non-negative, this implies for example that each g(p) is non-negative. But each g(n) is a product of terms of the form g(p), hence is also non-negative.
    Okay, this would work, but it presupposes that F(s) can be written in the form \prod_p(1-a_p p^{-s})^{-1} 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

    g(p^m) = (1+\chi_1(p^m))(1+\chi_2(p^m))

    so g(p) = (1+\chi_1(p))(1+\chi_2(p))

    but then g(p)^m = (1+\chi_1(p))^m (1+\chi_2(p))^m \neq (1+\chi_1(p^m))(1+\chi_2(p^m)) = g(p^m).

    so the coefficients are actually not completely multiplicative.
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    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?
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  5. #5
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    Quote Originally Posted by Petek View Post
    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?
    Damn right it does! Much better, thank you .
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