Weil positivity and trace formula the archimedean place

The UV prolate spectrum matches the zeros of zeta

“…The prolate spheroidal wave functions play a key role in refs. 1 – 3 in relation to the Riemann zeta function. In all these applications, they appear as eigenfunctions of the angle operator between two orthogonal projections in the Hilbert space of even square integrable function on .…”

“…In ref. 2 , the compression of f ( S ) to Sonin’s space ( 11 ) (which consists of functions such that was shown to be (for λ = 1) the root of Weil’s positivity at the Archimedean place on test functions supported in the interval , but, since Sonin’s space is not preserved by scaling, one could not restrict scaling to this space. It turns out that commutes with the orthogonal projection on Sonin’s space.…”

Aspects of the screw function corresponding to the Riemann zeta‐function

“…Yoshida proved without any assumption that the hermitian form ⟨⋅, ⋅⟩ 𝑊 is positive definite on 𝐾(𝑎) if 𝑎 > 0 is sufficiently small ([33, Lemma 2]). Connes-Consani [4] provides an operator theoretic conceptual reason for this result. Yoshida also proved that, for given 𝑎 0 > 0 and 𝜇 > 0, there exists 𝑁 ⩾ 0 such that ⟨𝜙, 𝜙⟩ 𝑊 ⩾ 𝜇‖𝜙‖ 𝐿 2 for every 𝜙 ∈ 𝐾 𝑁 (𝑎) and 0 < 𝑎 ⩽ 𝑎 0 ([33, Lemma 3]).…”

“…Yoshida proved without any assumption that the hermitian form $$\langle \cdot ,\cdot \rangle _W$$ is positive definite on $$K(a)$$ if $$a&gt;0$$ is sufficiently small ([33, Lemma 2]). Connes–Consani [4] provides an operator theoretic conceptual reason for this result. Yoshida also proved that, for given $$a_0&gt;0$$ and $$\mu &gt;0$$, there exists $$N \geqslant 0$$ such that $$\langle \phi ,\phi \rangle _W \geqslant \mu \Vert \phi \Vert _{L^2}$$ for every $$\phi \in K_N(a)$$ and $$0&lt;a \leqslant a_0$$ ([33, Lemma 3]).…”

“…4 𝑒 −4𝑡 − 3𝜋𝑛 2 𝑒 −2𝑡 ) exp(−𝜋𝑛 2 𝑒 −2𝑡 ).Then 𝜉(1∕2 − 𝑖𝑧) = η(𝑧) for all 𝑧 ∈ ℂ ([29, §10.1]). In addition, −𝑖𝑧 𝜉(1∕2 − 𝑖𝑧) = η′ (𝑧) for all 𝑧 ∈ ℂ, since 𝜂(𝑡) is smooth and decays faster than any exponential as |𝑡| → +∞.…”