# Is it true that sum of reciprocal of primes $p$ such that $p|a_{f}(p)$ converges?

Let $$g(x)$$ be a polynomial with integral coefficients.

For $$r\geq 1$$, We define the sequence $$a_{g}$$ for some polynomial $$g(x)$$ as follows:

$$\clubsuit)a_{g}(1)=g(x)$$

$$\clubsuit)a_{g}(r)=g(a_{g}(r-1))$$ for $$r\geq 2$$

Now we are given a polynomial $$f(x)$$ of $$\deg(f)\geq 2$$ such that if $$x=0$$, $$a_{f}(r)\rightarrow\infty$$ as $$r$$ grows large.

Then is it true that if $$x=0$$ then sum of reciprocal of primes $$p$$ such that $$p|a_{f}(p)$$ converges?

P.S.An analogous result has been proven for elliptic curves by Serre.

• $a_g(2)=g(a_g(1))=g(g(x))$. $a_g(3)=g(a_g(2))=g(g(g(x)))$. So isn't $a_g(r)$ just $g^{(r)}(x)$, the $r$th power of $g$ under composition? Which is going to grow real fast, even for something as simple as $g(x)=x^2$? Mar 2, 2021 at 4:05
• Yes @above it would have a growth rate of c^{d^n}} Mar 2, 2021 at 4:11

Have you seen Silverman (Section 4)? The sum that he estimates there is the dynamical analogue of the relevant sums that appear when studying the analogue of $$\gcd(n,f^n(0))$$ (as in Kim for EDS, Sanna and myself for Lucas sequences... I am guessing this based on your post history). Otherwise I do not know about the exact sum you need, and I strongly suspect that such a bound is not in the literature - a search in the papers that cite Serre or Silverman turned out no results.
• I am afraid that's as close as it gets, the result of Serre is very classical but analogues for iterates are not studied much yet. Do note that the dynamical version of the problem introduces some genuinely new difficulty because of the growth of the iterates; the fibre bound that in the Lucas case gives a factor $1/2$ somewhere becomes $1/3$ in the EDS case because of growth, and becomes too large to be useful in the dynamical case; if you run some experiments you see that indeed several $p$ have very small $z(p)$. Good luck for your work though!