Continuations and Functional Equations
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چکیده
The first part of this writeup gives Riemann’s argument that the completion of zeta, Z(s) = π−s/2Γ(s/2)ζ(s), Re(s) > 1 has a meromorphic continuation to the full s-plane, analytic except for simple poles at s = 0 and s = 1, and the continuation satisfies the functional equation Z(s) = Z(1− s), s ∈ C. The continuation is no longer defined by the sum. Instead, it is defined by a wellbehaved integral-with-parameter. Essentially the same ideas apply to Dirichlet L-functions,
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