The H-EF bridge: zero-side and prime-side matrices agree

Let $Z$ be an abstract zero configuration, $P$ a choice of fixed parameters (taper profile $\varrho$, exponent $\lambda$ with $X = (T/2\pi)^\lambda$, ramp width $w$), and $T$ a height. With $L = \lambda \log(T/2\pi)$, taper $\varphi(u) = \varrho((L/2 - |u|)/w)$, grid points $\tau_k = T + 2\pi k/L$ and $d = \lfloor LT/2\pi \rfloor$, the paper's matrix $G$ [eq:Gdef] has two expressions:

• the zero side $G^{\mathrm z}_{kl} = \sum_\rho m_\rho\, \hat\varphi(\gamma_\rho - \tau_k)\, \hat\varphi(\gamma_\rho - \tau_l)$ (a tsum over all distinct zeros of $Z$; Z.Gz P T), and
• the prime side $G^{\mathrm p}_{kl} = \int_{\mathbb{R}} \hat\varphi(\tau - \tau_k)\, \hat\varphi(\tau - \tau_l)\, \nu_X(\tau)\, d\tau$ with $X = e^L$ (P.Gp T).

Statement. Assume H-EF (ExplicitFormulaPaper Z: the paper-form Weil explicit formula $W(f, g) = \int h_f \overline{h_g}\, \nu_X$ for all $C^2$ test functions supported in $[-L/2, L/2]$, together with its summability and integrability clauses), that $L(T) > 0$, that $\varphi \in C^2$ (as a complex-valued function), and that $\operatorname{supp} \varphi \subseteq [-L/2, L/2]$. Then

i.e. the two $d \times d$ matrices are equal entrywise. The proof applies H-EF to the test pair $f_k, f_l$ where $f_k(u) = \varphi(u) e^{-i\tau_k u}$, using $h_{f_k}(z) = \hat\varphi(z - \tau_k)$ and the realness of $\hat\varphi$ on $\mathbb{R}$.

Role. In the module Zeta23.Hypotheses.GzGp this bridge lets the matrix-variational argument evaluate the Gram matrix $G$ on the prime side; it is consumed by Zeta23.eventually_side_conditions on the way to Theorem A.