Distributions and Analytic Continuation of Dirichlet Series
نویسندگان
چکیده
Dirichlet series and Fourier series can both be used to encode sequences of complex numbers an , n ∈ N. Dirichlet series do so in a manner adapted to the multiplicative structure of N, whereas Fourier series reflect the additive structure of N. Formally at least, the Mellin transform relates these two ways of representing sequences. In this paper, we make sense of the Mellin transform of periodic distributions and other tempered distributions, as a tool for the analytic continuation of various L-functions and the derivation of functional equations. To illustrate what we mean, we sketch a heuristic argument for the functional equation of the Riemann zeta function. We let δn(x) denote the Dirac delta function at the point n ∈ Z. The sum Pn∈Z δn(x) is a tempered distribution; as such, it has a Fourier transform: (1.1) F¡ X n∈Z δn(x) ¢ = X n∈Z e(nx) ¡ e(x) =def e 2πix ¢ .
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