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I'll admit from the outset that this question is slightly vague. The actual question appears at the end of the post.

The explicit formula of Guinand and Weil can be written in the following way:

For 'nice' g (i.e. in $C_c^\infty(\mathbb{R})$)

$$\sum_\gamma \hat{g}(\gamma/2\pi) - \int_\mathbb{R} \frac{\Omega(\xi)}{2\pi}\hat{g}(\xi/2\pi) d\xi = \int_\mathbb{R} [g(x)+g(-x)] e^{-x/2}d(e^x-\psi(e^x)),$$ (1)

where the sum is over those $\gamma$ such that $1/2+i\gamma$ a non-trivial zero of the Riemann Zeta function, $\psi(x) = \sum_{n\leq x} \Lambda(n)$ is the Chebyshev prime counting function, and $$\Omega(\xi) = \tfrac{1}{2}\tfrac{\Gamma'}{\Gamma}(1/4+i\xi/2) + \tfrac{1}{2}\tfrac{\Gamma'}{\Gamma}(1/4-i\xi/2) - \log \pi.$$

Here $\gamma$ can possibly be complex.

It is usually proven using a contour integral to capture the zeroes of the Zeta function, then evaluating the integral a different way, making use of the reflection formula along with the arithmetical meaning of $\zeta(s)$ for $\Re s > 1$. (See for instance Montgomery and Vaughan.Vaughan, Multiplicative Number Theory.)

$\Omega(\xi)/2\pi \sim \log \xi /2\pi$, and is the mean density for the number of zeroes to occur in the critical strip with real part $\xi$. On the assumption of the Riemann hypothesis, the left hand side takes the nice form: $$\int_\mathbb{R}\hat{g}(\xi/2\pi)\bigg(\sum_\gamma \delta(\xi-\gamma) - \frac{\Omega(\xi)}{2\pi}\bigg) d\xi.$$

The explicit formula therefore expresses a Fourier duality between the error term in the Chebyshev prime counting function and the error term in the zero counting function. The structural reason why this duality arises is not really apparent to me from the contour integral proof above, and is what I'm really getting at with this question.

That said, left at this the question is a little imprecise, and there is something of a lie here because the form of the explicit formula where this becomes apparent involves assuming the Riemann hypothesis. Therefore:

Question: Is there a way not making use of entire function theory proper to show that there exist numbers $\gamma$ with $|\Im \gamma| \leq 1/2$, so that (1) is true?

A proof using harmonic analysis over the adeles would get bonus points.

One reason to be interested in a question like this beyond what I've elaborated above is to ask to what extent explicit formulas like (1) can be replicated for the 'Beurling primes.'

1

# The Guinand-Weil explicit formula without entire function theory

I'll admit from the outset that this question is slightly vague. The actual question appears at the end of the post.

The explicit formula of Guinand and Weil can be written in the following way:

For 'nice' g (i.e. in $C_c^\infty(\mathbb{R})$)

$$\sum_\gamma \hat{g}(\gamma/2\pi) - \int_\mathbb{R} \frac{\Omega(\xi)}{2\pi}\hat{g}(\xi/2\pi) d\xi = \int_\mathbb{R} [g(x)+g(-x)] e^{-x/2}d(e^x-\psi(e^x)),$$ (1)

where the sum is over those $\gamma$ such that $1/2+i\gamma$ a non-trivial zero of the Riemann Zeta function, $\psi(x) = \sum_{n\leq x} \Lambda(n)$ is the Chebyshev prime counting function, and $$\Omega(\xi) = \tfrac{1}{2}\tfrac{\Gamma'}{\Gamma}(1/4+i\xi/2) + \tfrac{1}{2}\tfrac{\Gamma'}{\Gamma}(1/4-i\xi/2) - \log \pi.$$

Here $\gamma$ can possibly be complex.

It is usually proven using a contour integral to capture the zeroes of the Zeta function, then evaluating the integral a different way, making use of the reflection formula along with the arithmetical meaning of $\zeta(s)$ for $\Re s > 1$. (See for instance Montgomery and Vaughan.)

$\Omega(\xi)/2\pi \sim \log \xi /2\pi$, and is the mean density for the number of zeroes to occur in the critical strip with real part $\xi$. On the assumption of the Riemann hypothesis, the left hand side takes the nice form: $$\int_\mathbb{R}\hat{g}(\xi/2\pi)\bigg(\sum_\gamma \delta(\xi-\gamma) - \frac{\Omega(\xi)}{2\pi}\bigg) d\xi.$$

The explicit formula therefore expresses a Fourier duality between the error term in the Chebyshev prime counting function and the error term in the zero counting function. The structural reason why this duality arises is not really apparent to me from the contour integral proof above, and is what I'm really getting at with this question.

That said, left at this the question is a little imprecise, and there is something of a lie here because the form of the explicit formula where this becomes apparent involves assuming the Riemann hypothesis. Therefore:

Question: Is there a way not making use of entire function theory proper to show that there exist numbers $\gamma$ with $|\Im \gamma| \leq 1/2$, so that (1) is true?

A proof using harmonic analysis over the adeles would get bonus points.

One reason to be interested in a question like this beyond what I've elaborated above is to ask to what extent explicit formulas like (1) can be replicated for the 'Beurling primes.'