On some explicit results for the sum of unitary divisor function

Speaker: Elchin Hasanalizade

Date: Thu, Oct 5, 2023

Location: PIMS, University of British Columbia, Online

Conference: UBC Number Theory Seminar

Subject: Mathematics, Number Theory

Class: Scientific


Let $\sigma^*(n)$ be the sum of all unitary (i.e. coprime) divisors of $n$. As an analogue of Lehmer’s totient problem, Subbarao proposed the following conjecture. The congruence $\sigma^*(n)\equiv 1\pmod{n}$ is possible iff $n$ is a prime power. This problem is still open. We strengthen considerably the lower estimations for the potential counterexamples to Subbarao’s conjecture.

In the second part of our talk, we discuss the growth of the function $\sigma^*(n)$. We establish a new explicit upper bound, namely $\sigma^*(n)<1.2678n\log\log{n}$ for all $n\ge223092870$. For this purpose, we use explicit estimates for Chebyshev’s $\theta$-function and for some product defined over prime numbers.