An attempt to interpret the Weil explicit formula from Beurling's spectral theory
Let \(\rho=\beta+i\gamma\) be the non-trivial zeros of the Riemann zeta-function. For certain functions \(F\) and for \(0<u<1,\nu\in{\mathbb R}\) the authors prove that \[ \lim_{T\to\infty}\sum_{|\gamma|<T} e^{\nu(\rho-1/2)}\int_{-\infty}^\infty F(\tfrac xu)e^{x(\rho-1/2)}\,dx=M(u,\nu)+R(u,\nu), \] where \[ M(u,\nu)=\begin{cases} 0,&\nu\neq \pm \log m, m=1,2,3,\dots\\ -\frac{\Lambda(m)}{m^{\frac 12}}F(0),&\nu=\pm \log m, m=2,3,4,\dots\\ -F(0)\log(2\pi u)-\int_0^\infty\left(\frac{F(x)+F(-x)}{2}-F(0)e^{-x}\right)\,\frac{dx}{x},&\nu=0 \end{cases} \] and \[ R(u,\nu)=O(u),u\to 0. \] If the Riemann hypothesis is true, then \[ R(u,0)=-2u\int_0^\infty\left(N(\lambda)-\frac{\lambda}{2\pi}\log\frac{\lambda}{2\pi}+\frac{\lambda}{2\pi}\right) \frac{d}{d\lambda}\left(\frac{\hat{F}(-u\lambda)+\hat{F}(u\lambda)}{2}\right)\,d\lambda, \] where \(N(\lambda)\) is the number of \(\rho=\frac 12+i\gamma\) in the range \(0<\gamma<\lambda\) counted with multiplicity, and \(\hat{F}\) is the Fourier transform of \(F\).
scientific article; zbMATH DE number 5859974
scientific article