2024/12/01 by Yuan Qiu, Qiu, Yuan, Alexander Kalmynin +1
Mathematics · #Functional Equations Stability Results #Fractional Differential Equations Solutions #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2412.00723
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(√(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 Ω-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 Ω-bound for the error term of summation of fractional sigma function σα(n) on the range 0 < α< (1)/(2), obtaining the result Ω((x ln x)^(1)/(4)+\fracα2).