Seungki Kim’s research while affiliated with University of Cincinnati and other places

We use state-of-art lattice algorithms to improve the upper bound on the lowest counterexample to the Mertens conjecture to $\approx \exp(1.96 \times 10^{19})$, which is significantly below the conjectured value of $\approx \exp(5.15 \times 10^{23})$ by Kotnik and van de Lune [KvdL04].

We use state-of-art lattice algorithms to improve the upper bound on the lowest counterexample to the Mertens conjecture to $\approx \exp (1.96 \times 10^{19})$, which is significantly below the conjectured value of $\approx \exp (5.15 \times 10^{23})$ by Kotnik and van de Lune (Exp Math 13:473–481, 2004).

We prove an estimate on the number of rational points on the Grassmannian variety of bounded twisted height, refining the classical results of Schmidt (Duke Math J 35:327–339, 1968) and Thunder (Compos Math 88(2):155–186, 1993) over the rational field: most importantly, our formula counts all points. Among the consequences are a couple of new implications on the classical subject of counting rational points on flag varieties.

We present extensions of the Siegel integral formula ([13]), which counts the vectors of the random lattice, to the context of counting its sublattices and flags. Perhaps surprisingly, it turns out that many quantities of interest diverge to infinity.