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In his plenary lecture "L-functions and Automorphic Representations" at the Seoul ICM James Arthur makes the following remark (Proceedings of the ICM 2014, vol. 1, p. 173):

Riemann conjectured that the only zeros of L(s) lie on the vertical line Re(s) = 1/2. This is the famous Riemann hypothesis, regarded by many as the most important unsolved problem in mathematics. Its interest stems from the fact that the zeros {ρ = 1/2 + it} of L(s) on this line are in some sense dual to prime numbers, or more accurately, to logarithms {γ = log pn} of prime powers. We can think of the former as a set of spectral data and the latter as a set of geometric data, which are related to each other by a Fourier transform.

Here $L(s)$ is the completed Riemann zeta function, so $L(s) = L(1-s)$.

(1) What is the "Fourier transform" that relates the primes ("geometric data") to the zeros of $L$ ("spectral data")?

(2) Is there a general Fourier transform relating points of a scheme over ${\mathbb{Z}}$ (resp. motive over $\mathbb{Q}$) to the zeros of its zeta function (resp. motivic $L$-function) of which this is an instance?

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This follows simply from contour integration. See https://en.wikipedia.org/wiki/Explicit_formulae_for_L-functions

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    $\begingroup$ thanks for the link! This answers question (1), in fact for zeta functions of arbitrary number fields and Hecke characters. Do you know of any higher-dimensional generalizations, my question (2)? $\endgroup$
    – Shun
    Commented Aug 26, 2020 at 5:29
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    $\begingroup$ This explicit formula can be generalised to hold for any $L$-function of an automorphic representation over $\mathrm{GL}_n(\mathbb{A}_F)$ for a number field $F$. $\endgroup$
    – Peter Humphries
    Commented Aug 26, 2020 at 10:20
  • $\begingroup$ @PeterHumphries: Thanks! Can you point to a reference? $\endgroup$
    – Shun
    Commented Aug 27, 2020 at 22:10
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    $\begingroup$ I'm not sure if it's written down anywhere, but the proof goes through unchanged, since all that is required is a functional equation, an Euler product, and control on the growth of the $L$-function (which follows from Stirling's formula and the Phragmen-Lindelof convexity principle). $\endgroup$
    – Peter Humphries
    Commented Aug 27, 2020 at 22:43
  • $\begingroup$ Isn't some form of the general case in the Iwaniec-Kowalski book on Analytic Number Theory? $\endgroup$
    – paul garrett
    Commented Aug 27, 2020 at 23:28

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