On common values of phi(n) and sigma(n), I

TL;DR

This paper demonstrates that, assuming a uniform prime k-tuples conjecture, there are infinitely many numbers up to x that are simultaneously in the ranges of Euler's totient function and the sum-of-divisors function, with their count approximately proportional to x divided by log x.

Contribution

It establishes a conditional asymptotic lower bound on the number of common values of phi(n) and sigma(n) under a conjecture related to prime tuples.

Findings

Conditional proof of infinitely many common values of phi(n) and sigma(n).

Asymptotic estimate for the count of such common values.

Relies on a uniform version of the prime k-tuples conjecture.

Abstract

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x(log x)^{-1+o(1)} numbers not exceeding x common to the ranges of Euler's function phi(n) and the sum-of-divisors function sigma(m).