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In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform Z (s) , s = σ + i t , with fixed 1 / 2 < σ < 1 , of the square | ζ (1 / 2 + i t) | 2 of the Riemann zeta-function. We consider probability measures defined by means of Z (σ + i φ (t)) , where φ (t) , t ⩾ t 0 > 0 , is an increasing to + ∞ differentiable function with monotonically decreasing derivative φ ′ (t) satisfying a certain normalizing estimate related to the mean square of the function Z (σ + i φ (t)) . This allows us to extend the distribution laws for Z (s) . [ABSTRACT FROM AUTHOR]

In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts (ζ (s + i t k α 1) , ... , ζ (s + i t k α r)) of the Riemann zeta function. Here, { t k : k ∈ N } is the sequence of Gram numbers, and α 1 , ... , α r are different positive numbers not exceeding 1. It is proved that the above set of shifts in the interval [ N , N + M ] , here M = o (N) as N → ∞ , has a positive lower density. For the proof, a joint limit theorem in short intervals for weakly convergent probability measures is applied. [ABSTRACT FROM AUTHOR]