# Distribution of residue classes of totients of (univariate) polynomials

Let $\phi$ denote Euler's totient function and $f$ a non-constant univariate polynomial with integer coefficients such that $f(u) \in \mathbf N^+$ for all $u \in \mathbf N^+$ (assume $f$ is irreducible if useful/necessary).

Q1. Is it true that $\mathfrak i_f := \inf_{n \ge 1} \frac{1}{n}|\phi(f(u)) \bmod n: u \in \mathbf N^+\}| > 0$?

We have from  that if a residue class (modulo a fixed $n \in \mathbf N^+$) contains at least one multiple of 4 then it contains infinitely many totients (i.e., stuff in the image of $\phi$), which implies that the answer to Q1 is affirmative if $f$ is linear and monic. In fact, I would be already happy to hear about other particular cases. So let me add the following:

Q2. What about the case when $f$ is linear/quadratic?

My motivation (for those here who dislike random questions) is somehow related to the study of the "density" (whatever this means) of sets of totients.

Bibliography.

 T. Dence and C. Pomerance, Euler's Function in Residue Classes, Ramanujan J. 2 (1998), 7-20 (click).

• I doubt you can prove something like that unconditionally. – sergey Jul 19 '15 at 14:29
• The answer, if positive, would resolve a "long standing" math.stack question math.stackexchange.com/questions/43020/… – sergey Jul 19 '15 at 14:37
• I doubt it too, but special cases (I mean, cases corresponding to some specific choice of $f$) would already be interesting to me. E.g., what about the linear/quadratic case? (Btw, thanks for the link.) – Salvo Tringali Jul 19 '15 at 14:42