How composite $a^n+b$ is?

Question

In connection with this question and its follow-up.

Suppose that $a\ge 2$ and $b\ne 0$ are integers, and $f$ is a monotonically increasing function such that $f(2)>1$, $f(p)\to\infty$, and the series $\sum_p 1/f(p)$ (extended onto all rational primes) diverges. Suppose further that for any integer $K>0$, there exist infinitely many integers $n\ge 1$ with $\gcd(a^n+b,K)=1$.

Is it true that, under the stated assumptions, every point of the interval $[0,1]$ is a limit point of the sequence with the $n$th term $$ \prod_{p\mid a^n+b}\Big(1-\frac1{f(p)}\Big)? $$ In particular, is it true that every point of $[0,1]$ is a limit point of the sequence $\varphi(2^n-1)/(2^n-1)$?

Answer 1

This is an answer to the part of the question where the OP is asking if the image of the function $$R_{2,\varphi}: \mathbf N^+ \to \mathbf R: n \mapsto \frac{\varphi(2^n - 1)}{2^n-1}$$ is dense in the interval $[0,1]$. The short answer is yes, some more details follow.

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This week, I met Carlo Sanna in Turin during a meeting of number theory and mentioned the problem to him. He remembered to have read about the same question in a paper of Florian Luca. Today, I got an e-mail from Carlo with the full reference:

F. Luca, On the sum of divisors of the Mersenne numbers, Math. Slovaca 53, No. 5 (2003), 457-466 (EuDML link).

More precisely, see point ii) of the theorem on the bottom of p. 458.

By the way, Luca points out on p. 459 that:

• his method works for any multiplicative function $f: \mathbf N^+ \to \mathbf R$ for which "there exist [...] $c > 0$ and $\lambda > 1$ so that $f(p^a) = 1 + \frac{c}{p} + O(p^{-\lambda})$ holds for all prime numbers $p$ and all positive integers $a$", on condition that $[0,1]$ is replaced with the interval $[\liminf_n f(n), \limsup_n f(n)]$ (the question in the OP corresponds to the case when $f = R_{2,\varphi}$);

• the conclusion remains true if the sequence of Mersenne numbers is replaced with a Lucas sequence subjected to some technical conditions.

Among other things, Luca's proof makes use of the Siegel-Walfisz theorem.