Arithmetical function and distributions

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Main Question or Discussion Point

can any Arithmetical function [tex] A(x)= \sum_{n\le x}a(n) [/tex]

be regarded as the train of dirac delta functions (its derivative)

[tex] dA = \sum_{n=1}^{\infty}a(n)\delta (x-n) [/tex]

from this definition could we regard the explicit formulae for chebyshev function

[tex] d\Psi(x) =1- \sum_{\rho}x^{\rho -1}- (x^{3}-x)^{-1} [/tex]

and from this, using the definition of Mellin transform, we could obtain the sums over the Riemann zeros for lots of function f(x) provided its Mellin transform exists.
 

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