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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 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?

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?

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 $L(s) = L(1-s)$.

What is the "Fourier transform" that relates the primes ("geometric data") to the zeros of $L$ ("spectral data")? Is there a general Fourier transform relating a motive over $\mathbb{Q}$ to the zeros of its motivic $L$-function of which this is an instance?

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 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?

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 $L(s) = L(1-s)$.

What is the "Fourier transform" that relates the primes ("geometric data") to the zeros of $L$ ("spectral data")? Is this an instance of a general Fourier transform relating a motive over $\mathbb{Q}$ to the zeros of its motivic $L$-function?

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 $L(s) = L(1-s)$.

What is the "Fourier transform" that relates the primes ("geometric data") to the zeros of $L$ ("spectral data")? Is there a general Fourier transform relating a motive over $\mathbb{Q}$ to the zeros of its motivic $L$-function of which this is an instance?