Value-Distribution of the Riemann Zeta-Function Along Its Julia Lines

For an arbitrary complex number $$a\ne 0$$ a ≠ 0 we consider the distribution of values of the Riemann zeta-function $$\zeta $$ ζ at the a-points of the function $$\Delta $$ Δ which appears in the functional equation $$\zeta (s)=\Delta (s)\zeta (1-s)$$ ζ ( s ) = Δ ( s ) ζ ( 1 - s ) . These a-points $$\delta _a$$ δ a are clustered around the critical line $$1/2+i\mathbb {R}$$ 1 / 2 + i R which happens to be a Julia line for the essential singularity of $$\zeta $$ ζ at infinity. We observe a remarkable average behaviour for the sequence of values $$\zeta (\delta _a)$$ ζ ( δ a ) .