The Li criterion and the Riemann hypothesis for the Selberg class. II (Q963009)

Let \(F\) be a function in Selberg's class. It satisfies a functional equation of the type: \[ \phi_F(s)=\omega \bar{\phi_F}(1-s), \] where \[ \phi_F(s)=F(s)Q_F^s\prod_{j=1}^r\Gamma(\lambda_j s+\mu_j), \] with \(Q_F>0\), \(\lambda_j>0\), \(\text{Re}\mu_j\geq 0\) and \(|\omega|=1\). For \(n\geq 1\) let \[ \lambda_F(n):=\sum_{\rho}\left[1-\left(1-\frac{1}{\rho}\right)^n\right], \] where the sum is over the non-trivial zeros of \(F(s)\). The authors prove that the Riemann hypothesis for the function \(F(s)\) is equivalent to: \[ \lambda_F(n)=\frac{d_F}{2}n\log n+c_Fn+O(\sqrt{n}\log n)\quad n\to\infty, \] where \[ c_F= \frac{d_F}{2}(\gamma-1)+\frac12\log(\lambda Q_F^2),\quad \lambda=\prod_{j=1}^r\lambda_j^{2\lambda_j}, \] where \(d_F=2\sum_{j=1}^r\lambda_j\) and \(\gamma\) is Euler's constant.