Omega theorem for fractional sigma function

TL;DR

This paper establishes a new Omega-bound for the error term in the summation of fractional sigma functions, extending classical results in analytic number theory.

Contribution

It provides the first Omega-theorem bound for fractional sigma functions with $0<\alpha<\frac{1}{2}$, improving understanding of their error terms.

Findings

Derived an Omega-bound of $(x \ln x)^{\frac{1}{4}+\frac{\alpha}{2}}$ for fractional sigma functions.

Extended classical error term bounds to fractional cases in number theory.

Contributed to the sparse literature on Omega-theorems for sigma functions.

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 $\Omega$-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 $\Omega$-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 $\Omega((x \ln x)^{\frac{1}{4}+\frac{\alpha}{2}})$.