The values of the Riemann zeta-function on generalized arithmetic progressions
In the paper, the authors study the mean of the values of the zeta-function on some generalized arithmetic progressions on the critical line. More specifically, they consider the distribution of values of the zeta-function on generalized arithmetic progressions of the form $1/2+ i(\gamma+\delta_1m_1 +\dots +\delta_r m_r)$, where $\gamma$ is a real number, $\delta_1,\dots,\delta_r$ are positive real numbers, and $m_1, \dots , m_r$ are positive integers. It is seen that the mean of the values of the zeta-function exists on these generalized arithmetic progressions. The case of resonance here is when all $\delta_j$'s oscillate with the same logarithm $\Lambda$ in the denominators. It is also seen that, in the case of existence, the value of $\Lambda$ is uniquely determined. Otherwise, when there is no such an (integer) $\Lambda$, the values $\delta_1, \dots,\delta_r$ are said to be not in resonance. The main result of the paper is a theorem the authors prove on the behavior of the mean of the values of the zeta-function on this kind of generalized arithmetic progressions, distinguishing the two cases, namely the resonant and the non-resonant ones.
scientific article; zbMATH DE number 7005659