Omega theorem for fractional sigma function

Abstract

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 $\hat{I}\text{c◯}$-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 $\hat{I}\text{c◯}$-bound for the error term of summation of fractional sigma function $\ddot{I}{ƒ}_{\hat{I}}\pm (n)$ on the range $0<\hat{I}\pm <\frac{1}{2}$, obtaining the result $\hat{I}\text{c◯}((x\ln x{)}^{\frac{1}{4}+\frac{\hat{I}}{\pm }2})$.

13 pages