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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, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin transform approximates the identical zero in the strip { s ∈ C : 1 / 2 < σ < 1 } . [ABSTRACT FROM AUTHOR]

This paper is devoted to the approximation of a certain class of analytic functions by shifts Z (s + i τ) , τ ∈ R , of the modified Mellin transform Z (s) of the square of the Riemann zeta-function ζ (1 / 2 + i t) . More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts Z (s + i τ) , which approximate a given analytic function from F with a given accuracy. In the proof, the weak convergence of measures in the space of analytic functions is applied. Then, the set F coincides with the support of a limit measure. [ABSTRACT FROM AUTHOR]