An ergodic value distribution of certain meromorphic functions

TL;DR

This paper introduces a method to compute mean-values of meromorphic functions using ergodic transformations, with applications to zeta and L-functions, and relates it to the Lindelöf hypothesis.

Contribution

It develops a novel ergodic approach to analyze mean-values of meromorphic functions, including zeta and L-functions, and links these to the Lindelöf hypothesis.

Findings

Derived explicit formulas for mean-values using ergodic transformations.

Applied the method to zeta and L-functions.

Established an equivalence of the Lindelöf hypothesis with ergodic value distribution.

Abstract

We calculate a certain mean-value of meromorphic functions by using specific ergodic transformations, which we call affine Boolean transformations. We use Birkhoff's ergodic theorem to transform the mean-value into a computable integral which allows us to completely determine the mean-value of this ergodic type. As examples, we introduce some applications to zeta functions and $L$-functions. We also prove an equivalence of the Lindel\"{o}f hypothesis of the Riemann zeta function in terms of its certain ergodic value distribution associated with affine Boolean transformations.