Riemann-Type Functional Equations -- Julia Line and Counting Formulae --

Abstract

We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number $a\ne 0$ and a function from the Selberg class $\mathcal{L}$, we prove a Riemann-von Mangoldt formula for the number of a-points of the $\Delta$-factor of the functional equation of $\mathcal{L}$ and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these $a$-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of $\mathcal{L}(s)$ taken at these points.

28 pages, a part of the original version uploaded last year

References in corpus (3)

• On the value-distribution of the Riemann zeta-function on the critical line
• Value-Distribution of the Riemann Zeta-Function along its Julia Lines
• Classification of L-functions of degree 2 and conductor 1