On Weil's explicit formula (Q2516912)

From the summary and the introduction: ``A program of proving the Riemann hypothesis by using Fourier analysis on global fields is given by \textit{A. Connes} [Sel. Math., New Ser. 5, No. 1, 29--106 (1999; Zbl 0945.11015)]. The difficulty for realizing the program lies in proving the validity of Connes' global trace formula on an \(L^2\)-space. In this paper, a new global trace formula is obtained on a Fréchet space which gives the Weil distribution \(\Delta(h)\).'' ``In order to study the localization of nontrivial zeros of the Riemann zeta function we may have to consider Weil's explicit formula over global fields.'' ``By using Grothendieck's deep theory of nuclear locally convex spaces, we prove in this paper that \(\Delta(h)\) is equal to the spectral trace of a trace class operator on a complete nuclear locally convex space over global fields.'' This is a long and technical article. Section 2 includes results about some nuclear (Fréchet) locally convex spaces over global fields. Sections 3 to 7 present the trace evaluation of several trace class operators on different spaces.