Series of rationals

Question

Let $[a, b]$ be an interval in $\mathbb R$, does there exist a function $f(q)$ such that

$$\sum_{q\in [a, b]\cap \mathbb Q}f(q)<+\infty\ ?$$

Since there exists a bijection between $\mathbb N$ and $\mathbb Q$ and since $\sum_{n\in \mathbb N}\frac{1}{n^2}<\infty$, I think it is possible...

Do you know if it is possible to find such a function f(q)?

Answer 1

First enumerate the numbers in $[a,b]\cap \mathbb{Q}$ in whatever order, and call them $\{q_1,q_2,q_3,\ldots\}$. And then take $$\sum_{n=1}^{\infty}\frac{q_n}{n!}.$$

This series surely converges. We know that $$\sum_{n=1}^{\infty}\frac{2^n}{n!}$$ converges, and $|q_n|\geq|2^n|$ only for finitely many $n$, because we hold $[a,b]$ fixed.

Answer 2

If you want an explicit expression, go with $f\left({p\over q}\right)={1\over q^3}$, for example. The number of fractions with denominator $q$ on $[0,1]$ is less than $q$, so the sum over them is less than $1\over q^2$, so here we go. In this case you might even be able to find the sum explicitly: it seems to be $\zeta(2)\over\zeta(3)$.

The generalization to $[a,b]$ is obvious.