Warning! This set of interlinked pages is very much under construction. If you are already familiar with the subject matter, I would appreciate any help you might provide. If you are seeking information on the subject matter, I would be interested to hear what you would ultimately like this page to contain. Thank you.

 Gutzwiller Trace Formula ___  [link]   ___ Selberg Trace Formula | | [link] | | | | [link] | | Riemann's zeta function ___  [link]   ___ Riemann-Weil Explicit Formula | | [link] | | | | [link] | | Connes'(hypothetical) trace formula ___  [link]   ___ (Deninger's) Lefschetz Trace Formulae

This is fairly straightforward. It's basically a matter of generalisation.

Riemann put forward an explicit formula (proved rigorously by von Mangoldt) which relates a prime counting function and a sum over the zeros of the Riemann zeta function in C.

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 [1952] 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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