# MUIC Math

Mathematics at MUIC

### Internal

seminar:pp_floor_fn

# New Sums Involving Floor Function Introduced by Tverberg

Prapanpong Pongsriiam
Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom

16:00 Wednesday 3rd May 2017, Room 3303, MUIC

### Abstract

The floor function of $x$, denoted by $\left\lfloor x\right\rfloor$, is defined to be the largest integer less than or equal to $x$. In number theory, it is often desirable to approximate the sums which involve floor function. Tverberg has recently introduced new sums such as $$S_{a_1,a_2,a_3;m}(K) = \sum_{k=0}^{K}f_{a_1,a_2,a_3;m}(k),$$ where \begin{multline*} f_{a_1,a_2,a_3;m}(k) = \left\lfloor \frac{a_1+a_2+a_3+k}{m}\right\rfloor-\left\lfloor \frac{a_1+a_2+k}{m}\right\rfloor-\left\lfloor \frac{a_1+a_3+k}{m}\right\rfloor\\ -\left\lfloor \frac{a_2+a_3+k}{m}\right\rfloor+\left\lfloor \frac{a_1+k}{m}\right\rfloor+\left\lfloor \frac{a_2+k}{m}\right\rfloor+\left\lfloor \frac{a_3+k}{m}\right\rfloor-\left\lfloor \frac{k}{m}\right\rfloor. \end{multline*}

In this talk, I will show how Tverberg obtains upper and lower bounds of such sums. I will also give an idea on how to obtain upper and lower bounds for a more general sum. This is joint work with Kritkhajohn Onpeang, my master student.