Markov processes on the adeles and Chebyshev function
The author obtains a representation formula for the values of Chebyshev's function \(\psi(x)\) by the exit time \(T\) of adeles-valued processes from the finite integral adeles \(\mathbb{R}\times\prod_{p \in {\mathcal P}}\mathbb{Z}_p\), i.e., that there holds relation \(\psi(x)=\operatorname{E}(T(x))^{-1}\) (here \(\mathcal P\) is the set of all primes, \(\mathbb{R}\) is the set of all real numbers, \(\mathbb{Z}_p\) are the integer rings). From this relation, it is easy to see that the open assertion \(\zeta(s)\not =0\) if \(\operatorname{Re} s>\alpha\) for some \(\frac{1}{2} \leq \alpha <1\) can be described by the asymptotic behavior of the expectation of the exit times \(T(x)\) as \(x \to \infty\).
scientific article; zbMATH DE number 6130789