An interesting way of expressing any real number using the harmonic series.

Question

I recently saw the identity $$ \frac{1}{1} - {1 \over 2} +{1 \over 3} - {1 \over 4} + {1 \over 5} - {1 \over 6} \dotsb = \log(2) $$ which I found rather interesting. I was intrigued by the way a transcendental number like $\log(2)$ could be expressed by using the reciprocals of every integer once. I then wondered what other real numbers could be represented this way, by using the infinite harmonic series, but changing the $+$ or $-$ signs before each fraction.

I soon realized that every real number could be represented this way, although not necessarily with such a clear pattern to the signs. Since + and - are binary options, I realized that the pattern of $+$ or $-$'s in the series could be represented by using a binary decimal number between 0 and 0.11111...

I called this number the "harmonic sign number" (or "HSN" for short). For instance, an example of a harmonic sign number for $\log(2)$ would be $0.\overline{10}$, because it alternates between $+$ and $-$, with the 1's representing a + and the 0's representing a -. However, $\log(2)$ is probably a special case, as it has a harmonic sign number that is rational.

It seems like most numbers could only be represented by irrational harmonic sign numbers, however that is still up for debate.

Another example of a rational HSN would be $0.\overline{1100}$, which is the HSN for ${\pi \over 4} + {\log(2) \over 2}$

Also, every real number has an infinite amount of HSN's, but every HSN corresponds to only one real number (or not, if it doesn't converge).

Some questions that I have thought about:

1. Which numbers have rational harmonic sign numbers? Can only trancendental numbers have rational HSN's?
2. If not, is there a quick way to tell whether a number has a rational HSN?
3. Are there any rational numbers that have rational or at least algebraic HSN's?
4. Does there exist a number which is it's own HSN? Are there more than one? Infinite?
5. Which HSN's actually converge? I would suggest that they would have to be normal.

Please note, this is done purely out of interest, so only respond if you are actually interested! If you would like clarification also please ask in comments and I will edit. Also, I'm not looking for full concrete answers. Feel free to answer with anything you have found or is of interest. It does not need to answer the questions posed above.

Answer 1

Some notation first: $$\mbox{HSN}(z)=H_z=\{a_0,a_1,\dots\}=0.a_0a_1\dots, \; \forall a_i. |a_i|=1$$ $$ z = \sum_{i=0}^\infty{\frac{c_i}{i+1}}, \; c_i = \begin{cases} +1 & \mbox{if } a_i=1 \\ -1 & \mbox{if } a_i=0 \end{cases} $$

Now we can write $z$ as, $$(1) \Rightarrow z =\int_0^1P_z(x)dx,\; P_z(x) = {\sum_{i=0}^\infty{c_ix^i}dx}$$

Rational HSN

define $R_{z} = \{p\in \mathbb{N}_0|\forall a_i \in H_z. a_i=a_{i+p}\}$ and $r_z = min(R_z)$ then is $H_z$ rational $\Leftrightarrow R_z \ne \varnothing$

$$ \begin{align*} H_z \text{ is rational} \Rightarrow & P_z(x)&& = {\sum_{i=0}^\infty{c_ix^i}} \\ & && = {\sum_{i=0}^\infty{c_{(i\mbox{ mod }r_z)}x^i}} \\ & && = {\sum_{i=0}^{r_z-1}{c_ix^i}} + {\sum_{i=r_z}^\infty{c_{(i\mbox{ mod }r_z)}x^i}} \\ & && = {\sum_{i=0}^{r_z-1}{c_ix^i}} + x^{r_z}{\sum_{i-r_z=0}^\infty{c_{(i-r_z\mbox{ mod }r_z)}x^{i-r_z}}} \\ & && = p_z(x) + x^{r_z}P_z(x), \; p_z(x)={\sum_{i=0}^{r_z}{c_ix^i}}\\ \Leftrightarrow& P_z(x) && = \frac{p_z(x)}{1-x^{r_z}}\\ \Leftrightarrow& z && = \int_0^1\frac{p_z(x)}{1-x^{r_z}}dx\\ \end{align*}$$

convergence

$p_z(1) \neq 0$ will cause $\frac{p_z(x)}{1-x^{r_z}}$ to converge to $\frac1x$ which has infinite area in $[1-\epsilon,1]$. So in order for $z$ to have a finite value $ p_z(1) $ must be $0$, or $\sum_{i=0}^{r_z-1}{c_i} = 0$. This also means that $r_z$ must be even.

closed form

we can solve the integral for $z$ using the partial fraction decomposition of $\frac{p_z(x)}{1-x^{r_z}}$

$$\begin{align*} (2) \Rightarrow && \frac{p_z(x)}{1-x^{r_z}} & = \sum_{t=1}^{r_z}{\frac{p_z(\alpha_t)\alpha_t}{r_z(x-\alpha_t)}}, \; \alpha_t = e^{2\pi ti/{r_z}}\\ \Rightarrow && z &= \int_0^1{\sum_{t=1}^{r_z}{\frac{p_z(\alpha_t)\alpha_t}{r_z(x-\alpha_t)}}dx}\\ && z &= \sum_{t=1}^{r_z}{\frac{p_z(\alpha_t)\alpha_t}{r_z}\cdot\ln\left(\frac{1-\alpha_t}{0-\alpha_t}\right)} \\ && z &= \sum_{t=1}^{r_z}{\frac{p_z(\alpha_t)\alpha_t}{r_z}\cdot\ln(1-\bar\alpha_t)} \\ && z &= \frac{-p_z(-1)}{r_z}\cdot\ln(2) + \sum_{t=1}^{r_z/2-1}{\frac{p_z(\alpha_t)\alpha_t}{r_z}\cdot\ln(1-\bar\alpha_t)+\frac{p_z(\bar\alpha_t)\bar\alpha_t}{r_z}\cdot\ln(1-\alpha_t)} \\ && z &= \frac{-p_z(-1)}{r_z}\cdot\ln(2) + \sum_{t=1}^{r_z/2-1}{\frac{p_z(\alpha_t)\alpha_t}{r_z}\cdot\overline{\ln(1-\alpha_t)}+\frac{\overline{p_z(\alpha_t)\alpha_t}}{r_z}\cdot\ln(1-\alpha_t)} \\ && z &= \frac{-p_z(-1)}{r_z}\cdot\ln(2) + \sum_{t=1}^{r_z/2-1}{\frac{p_z(\alpha_t)\alpha_t}{r_z}\cdot\overline{\ln(1-\alpha_t)}+\overline{\frac{p_z(\alpha_t)\alpha_t}{r_z}\cdot\overline{\ln(1-\alpha_t)}}} \\ && z &= \frac{-p_z(-1)}{r_z}\cdot\ln(2) + \frac2{r_z}\sum_{t=1}^{r_z/2-1}{Re\left(p_z(\alpha_t)\alpha_t\cdot\overline{\ln(1-\alpha_t)}\right)} \\ \end{align*}$$ (!!) because we assumed $z$ is finite the partial fraction for $\alpha_0$ can be omitted

Question 1;

I suspect that only transcendental numbers have a rational HSN, because they can be writen as a sum of natural logaritm of algebraic numbers, so the sum of transcendental numbers. However not all sums of transcendental numbers are transcendental, you'd have to prove that ${\frac{p_z(\alpha_t)\alpha_t}{r_z}\cdot\ln(1-\bar\alpha_t)}$ are algebraically independent.

Question 2:

as far as i found not, the opposite would be easier (if Question 1)

Question 3:

If Question 1 is correct then no algebraic number could have a rational HSN

Question 4:

If Question 1 is correct, such a number would have to be irrational.

Question 5:

For rational HSN see above, for irrational HSN i suspect the same must be true.

$(1)$ $$\frac1n=\int_0^1{x^{n-1}dx}$$ $(2)$ $$\frac{P(x)}{Q(x)}=\sum_{i=0}^n{\frac{P(\alpha_i)}{Q'(\alpha_i)}\frac1{x-\alpha_i}}$$