5
$\begingroup$

Let $n\geq 1$ an integer, in this post I denote the product of distinct prime numbers dividing dividing $n$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ is the famous arithmetic function that appears in the abc conjecture. See it you want the Wikipedia Radical of an integer

Claim. It's easy to prove that the ordinary generating function $$f(z)=\sum_{n=1}^\infty \operatorname{rad}(n)z^n\tag{1}$$

for the square-free kernel or radica of $n$ has radius de convergence $1$.

Proof. It's obvious the inequality $1\leq \operatorname{rad}(n)\leq n$ thus $1\leq (\operatorname{rad}(n))^{1/n}\leq n^{1/n}$, and from here squeeze theorem implies $(\operatorname{rad}(n))^{1/n}\to 1$. And the Cauchy–Hadamard theorem that $R=1$.$\square$

Question 1. I'm curious about what standard claims/questions, if any, can be stated about the singularities of a the generating function $f(z)$ over $|z|=1$. That is, imagine that we want to study the singularities of $f(z)$ over $|z|=1$, what questions can be studied? Many thanks.

Thus I'm asking what standard questions should can be studied being those potentially interesting. Aren't required deduction, only are required some details about what topics/issues can be studied about the set of singularities of $f(z)$ on the set of complex numbers $|z|=1$. If you know it, or more advanced questions from the literature refer it answering this Question 1 as a reference request, and I try to find and read the theory about previous generating function and its singularities from the literature.

I wondered previous and next question searching information about the generating function of the greatest prime factor (that is a different arithmetic function) in Internet that I found [1].

Question 2 (Optional). I would like to know* the phase plot for previous ordinary generating function $$f(z)=\sum_{n=1}^\infty \operatorname{rad}(n)z^n$$ in the same spirit that is showed in the first plot of the section Greatest Prime Factor from Linas' Mathematical Art Gallery [1]. Can you provide us, or do you know it from the literature, the phase plot for $(1)$ over $|z|\leq 1$? Many thanks.

I would like to see the plot (feel free if you want to add some mathematical details about how calculate it) with the purpose to know it.

References:

[1] Greatest Prime Factor, from Linas' Mathematical Art Gallery, home page of Lina Vepstas (2016).

$\endgroup$
  • $\begingroup$ Sorry, my comment may not work, I deleted it. $\endgroup$ – i707107 Jul 14 '18 at 23:27
  • $\begingroup$ Many thanks any case for you attention and help @i707107 $\endgroup$ – user243301 Jul 15 '18 at 9:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy