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This is fairly straightforward. It's basically a matter of generalisation.
The Riemann-Weil Explicit Formula (1952) generalises this. Burnol points out that it generalises the Riemann-von Mangoldt formula to an identity between two distributions. He goes on:
"In fact Weil's aim in this regard seems to have been at first to express 'the most general' Riemann-like relation between primes and zeros. One can argue whether such a thing exists, and more strongly still whether it really has been achieved by Weil."
The formula in  applies to a general algebraic number field
K and the character chi. If we take K = Q
and chi = the trivial character (the simplest case) then it
reduces to a formula stated on the
Riemann-Weil page (the version which overtly resembles the
Selberg Trace Formula). this still involves
test functions (called F or g in various versions), so we
indeed have an identity of two distributions. Correctly choosing a
family of test functions will then collapse this general formula down to the
familiar Riemann-von Mangoldt formula.
trace formulae, explicit formulae and number theory page
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