zeros of L_4(s) on Re(s)=1?

Discussion in 'Math Research' started by marco72, Sep 14, 2009.

  1. marco72

    marco72 Guest

    It is a well-known fact that the zeta function has no zeros on the
    line Re(s)=1. Consider the Dirichlet L-series
    associates to the non-trivial mod 4 character
    L_4(s) = 1/1^s - 1/3^s + 1/5^s - 1/7^s + ...
    Is it still true that this function has no zeros on the line Re(s)=1?
    If so, why? Thanks.
    marco72, Sep 14, 2009
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  2. marco72

    JoeShipman Guest

    This is true for any Dirichlet L-series. Newman's proof is the
    slickest: consider the product Zm(s) of the L-functions for the non-
    trivial characters mod m. Suppose Zm(1+ia)=0. Then so is Zm(1-ia), and
    the function ((Zm(s))^2)Zm(s+ia)Zm(s-ia) is both real and entire
    (because the only possible pole at s=1 is balanced by the zeros at
    1+ia and 1-ia). But the Dirichlet series for this function has non-
    negative real coefficients (because by Euler factorization the log of
    each factor has non-negative real coefficients and exponentiation
    preserves this). It is a standard result that if an entire function
    has a Dirichlet series with non-negative coefficients, then that
    series is everywhere convergent. But it's easy to show that the
    Dirichlet series diverges at 0 by looking at the subseries of terms
    where n is a power of some prime.
    JoeShipman, Sep 15, 2009
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