I'll admit from the outset that this question is slightly vague. The actual question appears at the end of the post.

The [explicit formula of Guinand and Weil][1] can be written in the following way:

*For 'nice' g* (i.e. in $C_c^\infty(\mathbb{R})$)

$$
\sum_\gamma \hat{g}(\gamma/2\pi) - \int_\mathbb{R} \frac{\Omega(\xi)}{2\pi}\hat{g}(\xi/2\pi) d\xi
= \int_\mathbb{R} [g(x)+g(-x)] e^{-x/2}d(e^x-\psi(e^x)),
$$ (1)

*where the sum is over those* $\gamma$ *such that* $1/2+i\gamma$ *a non-trivial zero of the Riemann Zeta function,* $\psi(x) = \sum_{n\leq x} \Lambda(n)$ *is the Chebyshev prime counting function, and* 
$$\Omega(\xi) = \tfrac{1}{2}\tfrac{\Gamma'}{\Gamma}(1/4+i\xi/2) + \tfrac{1}{2}\tfrac{\Gamma'}{\Gamma}(1/4-i\xi/2) - \log \pi.
$$

Here $\gamma$ can possibly be complex. 

It is usually proven using a contour integral to capture the zeroes of the Zeta function, then  evaluating the integral a different way, making use of the reflection formula along with the arithmetical meaning of $\zeta(s)$ for $\Re s > 1$. (See for instance Montgomery and Vaughan.) 

$\Omega(\xi)/2\pi \sim \log \xi /2\pi$, and is the mean density for the number of zeroes to occur in the critical strip with real part $\xi$. On the assumption of the Riemann hypothesis, the left hand side takes the nice form:
$$
\int_\mathbb{R}\hat{g}(\xi/2\pi)\bigg(\sum_\gamma \delta(\xi-\gamma) - \frac{\Omega(\xi)}{2\pi}\bigg) d\xi.
$$

The explicit formula therefore expresses a Fourier duality between the error term in the Chebyshev prime counting function and the error term in the zero counting function. *The structural reason why this duality arises is not really apparent to me from the contour integral proof above, and is what I'm really getting at with this question.*

That said, left at this the question is a little imprecise, and there is something of a lie here because the form of the explicit formula where this becomes apparent involves assuming the Riemann hypothesis. Therefore:

**Question:** Is there a way not making use of entire function theory proper to show that there exist numbers $\gamma$ with $|\Im \gamma| \leq 1/2$, so that (1) is true?

A proof using harmonic analysis over the adeles would get bonus points.

One reason to be interested in a question like this beyond what I've elaborated above is to ask to what extent explicit formulas like (1) can be replicated for the 'Beurling primes.'

  [1]: http://en.wikipedia.org/wiki/Explicit_formula#Weil.27s_explicit_formula