Omega theorem for fractional sigma function

The research in the subfield of analytic number theory around error term of summation of sigma functions possesses a history which can be dated back to the mid-19th century when Dirichlet provided an $O(\sqrt{n})$ estimation of error term of summation of $d(n)$. Later, G. Voronoi, G. Kolesnik, and M.N. Huxley (to name just a few) contributed more on the upper bound on the error term of summation of sigma functions. As for $\mathrm{\Omega }$-theorems, G.H. Hardy was the first contributor. Later researchers on this topic include G.H. Hardy and T.H. Gronwall, but the amount of academic effort is much sparser than $O$-theorems. This research aims to provide a better $\mathrm{\Omega }$-bound for the error term of summation of fractional sigma function ${\sigma }_{\alpha }(n)$ on the range $0<\alpha <\frac{1}{2}$, obtaining the result $\mathrm{\Omega }((x\ln x{)}^{\frac{1}{4}+\frac{\alpha }{2}})$.