Arithmetical function and distributions

[zetafunction]

can any Arithmetical function $$A(x)= \sum_{n\le x}a(n)$$

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

$$dA = \sum_{n=1}^{\infty}a(n)\delta (x-n)$$

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

$$d\Psi(x) =1- \sum_{\rho}x^{\rho -1}- (x^{3}-x)^{-1}$$

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.

 

[fresh_42]

Here's paper about the relation to the Gamma function:
http://math.uchicago.edu/~may/REU2014/REUPapers/Duncan.pdf