Prolate spheroidal wave functions, Sonine spaces, and the Riemann zeta function
For any non-Archimedean valuation \(\nu\) of \(k\), the field of rational numbers, denote by \(k_\nu\) the completion of \(k\) at \(\nu\), by \(O_\nu\) its maximal compact subring and by \(P_\nu\) the unique maximal ideal of \(O_\nu\). Let \(\psi_\nu\) be the additive character on \(k_\nu\) and for each valuation \(\nu\) of \(k\) select a fixed Haar measure \(d\alpha_\nu\) on the additive group \(k_\nu\) which is the ordinary Lebesgue measure on the real line and for a finite \(\nu\), the measure for which \(O_\nu\) has measure \(1\). Let \(S\) be a finite set of valuations of \(k\) containing the Archimedean valuation \(\infty\) and let \(S' = S \setminus \{\infty\}\), \(\mathbf A_S = \mathbf R \times \prod_{\nu \in S'}k_\nu\) and \(O^*_S = \{\eta \in k^* : | \eta|_\nu = 1, \nu \not \in S\}\). For \(f=\prod_{\nu \in S}f_\nu \in L^2(\mathbf A_S)\), \(\psi_S(x) = \prod_{\nu\in S}\psi_\nu(x_\nu)\) and \(\beta = (\beta_\nu) \in \mathbf A_S\), define \[ H_Sf(\beta) = \int_{\mathbf A_S}f(\alpha)\psi_S(-\alpha\beta)d\alpha \] For \(X_S = \mathbf A_S/O^*_S\), define \(L^2(X_S)\) as the Hilbert space that is the completion of the Schwarz-Bruhat space \(S(\mathbf A_S)\) and by \(L_e^2(X_S)\) its subspace generated by all functions in \(S(\mathbf A_S)\) such that \[ f(x_{\nu_1}, \dots , x_{\nu_{j-1}}, -x_{\nu_j}, x_{\nu_{j+1}}, \dots, x_{\nu_n}) = f(x) \] for all \(x = (x_{\nu_1}, \dots, x_{\nu_n}) \in \mathbf A_S\) and for \(j = 1, \dots, n\). Denote by \(L_\Lambda\) the orthogonal projection of \(L^2_e(X_S)\) onto the subspace \[ L_\Lambda = \{f \in L^2_e(X_S): f(x) = 0 \;for \;| x|_S \geq \Lambda^{-1}\} \] and by \(Q_\Lambda\) the subspace of all functions \(f\) in \( L^2_e(X_S)\) such that \(H_Sf(\alpha)\) vanishes for \(| \alpha |_S < \Lambda^{-1}\). The corresponding orthogonal projection of \(L^2_e(X_S)\) onto \(Q_\Lambda\) is also denoted by \(Q_\Lambda\). This main results of this paper are: \newline 1. A reformulation of Weil's form of the explicit formula involving the non-trivial zeros of the Riemann zeta function. \newline 2. A reformulation of Bombieri's refinement of Weil's criterion. \newline 3. A reformulation of Weil's form of the explicit formula involving the non-trivial zeros of the Riemann zeta function, expressed as the trace of an operator acting on \(L^2(X_S)\). \newline 4. A formula for \(Q_\Lambda\), namely \(Q_\Lambda = 1- H_SL_\Lambda H_S\). \newline 5. An explicit formula for the orthogonal projection of \(L^2_e(X_S)\) onto the Sonine space \(B_\Lambda\). \newline
scientific article; zbMATH DE number 6009665
scientific article