The analytic properties of the Mellin transform of the second power of the “short” sum from the Riemann zeta-function approximate equation

The approximate functional equation for $\left|\zeta\left(\dfrac{1}{2}+it\right)\right|^{2}$ ($t\gg 1$) is a sum of two sums and remainder. The first sum, called a “short” sum, contains $O(t^{2\varepsilon})$ terms, and the second sum contains $O(t^{2(1-\varepsilon)})$ terms ($0<\varepsilon<\frac12$). In this paper, we study analytic properties of the Mellin transform of the second power of the “short” sum absolute value and compare them with the corresponding properties of the Mellin transform of $\left|\zeta\left(\dfrac{1}{2}+it\right)\right|^{4}$.