Arithmetical fucntion and distributions

1. Mar 27, 2010

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.