Eigenvalues of the Laplace-Beltrami operator and the von-Mangoldt function
Copyright policy )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.
- Rikkyo University Japan
logarithmic derivative of the Selberg zeta-function, Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas), Spectral theory; trace formulas (e.g., that of Selberg), logarithmic derivative of the Riemann zeta-function, \(\zeta (s)\) and \(L(s, \chi)\), von-Mangoldt function, spectrum of the Laplacian, Perron formula
logarithmic derivative of the Selberg zeta-function, Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas), Spectral theory; trace formulas (e.g., that of Selberg), logarithmic derivative of the Riemann zeta-function, \(\zeta (s)\) and \(L(s, \chi)\), von-Mangoldt function, spectrum of the Laplacian, Perron formula
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).2 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!