Eigenvalues of the Laplace-Beltrami operator and the von-Mangoldt function
A relation between the average \(\int_ 0^ X \sum_{n=1}^ y \Lambda(n) dy\) of the von-Mangoldt function \(\Lambda(n)\) and the spectrum of the Laplacian for \(L^ 2 (\Gamma \setminus{\mathcal H})\) with \(\Gamma\) the modular group is proved. The proof is based on a Perron formula for the logarithmic derivative of the Selberg zeta-function. The terms involving the zeros of the Riemann zeta-function can then be expressed as a (well-known) Perron integral for the logarithmic derivative of the Riemann zeta-function which produces the von-Mangoldt function.
logarithmic derivative of the Selberg zeta-function
logarithmic derivative of the Riemann zeta-function
scientific article; zbMATH DE number 559495