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On a representation of the idele class group related to primes and zeros of $$L$$-functions. (English) Zbl 1079.11044
In the study of zeta and $$L$$-functions of curves over finite fields a critical step is the identification of the “numerator” of the function with the characteristic polynomial of an operator. This is usually done either in terms of the Jacobian or of étale cohomology. Neither of these exists in the number-field case and it has been a major problem for something like 60 years to find such an interpretation of the Hadamard product representation of such a function. As was first indicated by A. Weil one can interpret the Hadamard product representation in terms of a distributional identity (“explicit formulæof prime number theory”).
There have been several interpretations of these formulæin such a fashion as to have a cohomological interpretation; see, for example, C. R. Matthews, “Spectral analysis of the action of ideles on adèles” [J. Lond. Math. Soc. (2) 32, 392–398 (1985; Zbl 0614.12009)], D. Goldfeld, “Explicit formulae as trace formula” [in: Number Theory, Trace Formulas and Discrete Groups, Symp. in honor of A. Selberg, Oslo, Norway, 281–288 (1987; Zbl 0668.10050)], A. Connes, “Trace formula in noncommutative geometry and the zeros of the Riemann zeta function” [Sel. Math., New Ser. 5, 29–106 (1999; Zbl 0945.11015)].
The paper under review is devoted to examing this interpretation in more detail. The problem is that the underlying space is $$K_{\mathbb A}/K^{\times}$$ where $$K$$ denotes a number field. This is very singular and it is difficult to understand the function spaces on it. It follows Connes’ conceptions but instead of using Hilbert spaces, which assume the validity of the generalized Riemann Hypothesis, the author uses nuclear bornological spaces. This is in many ways a more natural setting for Weil’s formula (and for a number of other contexts). The author shows how it can be used to reprove the prime number/ideal theorem without reference to the theory of Dirichlet series.

##### MSC:
 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 22D12 Other representations of locally compact groups 43A35 Positive definite functions on groups, semigroups, etc. 58B34 Noncommutative geometry (à la Connes)
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