SEMINAR:On the sum of the $k$-th powers of positive integers that are...

Guest: Doğa Can Sertbaş, İstinye University

Title: On the sum of the $k$-th powers of positive integers that are coprime to $n$

Date/Time: May 08, 2024, 13:40

Zoom: https://sabanciuniv.zoom.us/j/97093455905 

Abstract: (in LaTeX format) For given positive integers $n$ and $k$, the sum of the $k$-th powers of the first $n$ consecutive integers can be given as

$$

S_k(n)=1^k+\cdots +n^k=\sum_{m=1}^{n}m^k.

$$

Similarly, we define the sum of the $k$-th powers of the first $\varphi(n)$ positive integers that are coprime to $n$ as

$$

\varphi_k(n) = \sum_{\substack{a=1 \\ (a,n)=1}}^{n} a^k,

$$

where $\varphi(n)$ denotes the Euler-phi function. It is well-known that for all $n > 0$ the sum $S_1(n)$ divides $S_k(n)$ when $k$ is odd. Motivated by this result, in this talk we deal with the positive integer values of $k$ for which the sum $\varphi_1(n)$ divides $\varphi_k(n)$ for all $n > 0$. More generally, we define the set

$$

\mathcal D_s = \{s \le k : \varphi_s(n) \mid \varphi_k(n) \ \ \forall n \ge 1 \}.

$$

Using certain smooth numbers in short intervals together with the properties of Bernoulli numbers, we prove that for any given positive integer $s$, the set $\mathcal D_s$ is finite. Apart from that, we mention several properties related to the set $\mathcal D_1$. In particular, we find that the set $\mathcal D_1$ contains $\{1,3,15\}.$ With the aid of a computer algebra toolbox, we conclude that $ \mathcal D_1 \cap [1,5000] = \{1,3,15\}.$

Bio: After finishing his undergraduate studies at Izmir University of Economics, he graduated from the Mathematics Department at University of Bonn in 2015. He finished his PhD studies at Koc University in 2020, and became an assistant professor at Cukurova University in 2021. Since 2023, he has been a full-time assistant professor at the Mathematics Department of Istinye University. His research mainly focuses on computational aspects of number theory and cryptography.