Connected components of complex divisor functions

Abstract

For any complex number c, define the divisor function σc : N → C by σc(n) = d|n d c .Let σc(N) denote the topological closure of the range of σc. Extending previous work of the current author and Sanna, we prove that σc(N) has nonempty interior and has finitely many connected components if (c) ≤ 0 and c = 0. We end with some open problems.

On new arithmetic function relative to a fixed positive integer. Part 1

“…Many mathematicians have been studied the perfect numbers and their generalizations with the help of various arithmetic functions (see e.g., [7,8,11]). In [9,10], some arithmetic functions are used in characterizing generalized Mersenne primes.…”

On gaps in the closures of images of divisor functions

“…Because p m is r-mighty, it follows from Zubrilina's result that r > r pm . We know that p m ≥ q 2 ≥ 2 2 . By (1), this means that r pm ≥ r 5 .…”

“…where π(r) is the number of primes at most r. In addition, she showed that C r does not take on all finite values; in particular, she showed that C r = 4 for all real r. Such numbers are now called Zubrilina numbers [2].…”