We know that our target is going to have a non-terminating binary expansion if the denominator has a factor that is coprime with two. Let $a$ be a boolean value that is true iff $x$ has a factor that is coprime with two, false otherwise. If $a$, then let $b$ equal $\neg x$, else the result is $\lfloor log_2(x + 2) \rfloor$.
From here, it is obvious to see that $\sum_{n=1}^{\infty} (x + b)\cdot 2^{-n \lfloor log_2(x + 2)\rfloor } = 1$
Therefore, we have computed $\frac{1}{x}$ for $x \in \mathbb{N}$.
If we want to actually compute some portion of the reciprocal, we can use the negation of (the significant bits of) x and from there, determine how to sum to that value as a sum, $\sum_{n=1} x\cdot 2^{-n}$.
So... curious, has the Riemann zeta function been defined in terms of exp(z) yet? I mean, if we map Z(s) to exp(x + iy), then we can use complex argument to define an inverse by defining a map from exp(x + iy) to Z(s) and setting equal to zero for a given branch. Pretty sure that would solve the RH. Just an idea I had. No idea what's already been tried, either.
@copper.hat My reasoning is that it's probably easier to manipulate exp(x + iy) given the overwhelming information and many ways to manipulate it compared to Zeta(s). This could be said for any function really when defined fully in terms of exp and ln, and only elementary mathematical operations.
Not to mention, the two functions do look quite similar in their graphs already anyways as real functions.
If I had to begin somewhere, I'd probably begin by studying the gamma function and its relationship to $n^s$...
In that way, one could relate the terms of the infinite sum of Zeta(s) to the infinite sum of exp(z).
Suddenly makes things seem a lot more feasible, eh?
Anyways, I think that's all I have to say about the zeta function for now.
@copper.hat Basically the extract of dirt. You take rocks and you grind them into soil. Then you pour water over it at boiling temperatures and it creates a delicious beverage.
I have both a belgian coffee maker and an espresso machine. Once they're set up, I'm dumping keurig for good. The belgian one makes some of the best coffee I've ever had. Was introduced to it by my HS chemistry teacher.
we honestly have no idea what we're supposed to do. it doesn't help that legal english is informed by England who aped weird customs from france and latin and none of it makes any real sense.
i think that judges should have to wear wigs in this country.
i used to not care at all. then somehow when in the us, people thought i knew, so i made it up as i went along and left a trail of grammatical destruction behind.
i know enough to appear knowing, but i rely on common knowledge from those around me.
same here actually. now i am revealed as an imposter.
i found the last example in my inbox. we were talking about whether it should be sons of sam or son of sams, if we were referring to non-plural instances of a serial killer. i'll omit why we were talking about that.
i was a strong advocate for the first option because we were not talking about carbon copies of that one guy
If there were no absolutes, then nothing could be or exist as everything would ultimately then reduce to subjectivism which is a metaphysical impossibility. The subjective requires an objective for its existence.
Next necessarily relies on context. Context is the subjective element. In principle, next always refers to the object succeeding another object in a given context. The latter is an objective principle which defines next universally.
my wife's first introduction to my mother was my mother doing a version of the dead parrot sketch about a patient of hers who had just died. i'm not sure my wife had previously seen the dead parrot sketch. but we were both laughing.
my wife and i ran into my german teacher in a bakery about 10 years after i'd gone to school with him. he'd retired. the only thing he said to her was "have you had any luck with him?"
one thing i've generally noticed, and i don't mean to stereotype, is that women will be very more specific in their observations about people than men.
i've had my friends tell me wife stuff that i wouldn't tell a soul and vice versa.
men tend to be happy to be dumb idiots and that's the way i like it, because i'm a dumb idiot.
a friend of mine who knew me in high school talked to my wife at ben and nicks once when i went to the bathroom and i was like, what did you say to her, you can't let that out.
nobody talks about that.
they're friends on facebook now, who knows what they're discussing.
i've just never had a good experience with talking about a spouse of somebody so i just don't do it even if i could.
say nothing, watch, listen. be a pal. that's my motto.
wrote one test recently in which it was written something like this at the start that: you'll get 0 mark for every wrong answer, -2 for every unattempted answer etc.
