Developments in Proving that Euler Mascheroni constant is irrational.

[safe merlin]

Also is this anything worth publishing?

[patent stratus]

What exactly do you mean by "period of Euler's constant"?

[safe merlin]

The length of digits before the Euler Mascheroni constant starts repeating itself.

[patent stratus]

Do you have a proof of such thing?

[safe merlin]

The constant is conjectured to be transcendental. This limit is derived assuming it's rational.

So if this limit is not convergent then the constant is irrational.

[patent stratus]

I suggest you post this question on stack exchange. You will get better answers there I believe.

[glad adder]

Wdym publishing

Like for research or on stackexchange

[safe merlin]

For research

[safe merlin]

Yeah I think I should do that

[glad adder]

Id say that this topic isn't really heavy enough to constitute publishing since it's pretty much just a limit

[patent stratus]

I'd argue with that. Everybody starts somewhere, if you are certain that this has no flaws I encurage you to publish this. My first papers were all about pretty "uninteresting" topics as well but that's how you start a career.

[safe merlin]

Ok I'll take your argument into consideration

The reimann hyptheseis is also a limit problem btw the first time I found that out I was mind blown with how simply you can state it but how difficult it is.

[glad adder]

Well its not really just a limit but rather an entire class of functions with many different ways of observing it

[safe merlin]

Yes I do believe there are no errors. It's also very interesting to me that constant irrationality can be boiled down to disproving limit and the block of digits of ln 10 for any 10^n upto 10^n +n are random if we can prove they are random then the Euler Mascheroni constant would be transcendental.

[glad adder]

And I'd say it wouldn't be as famous as it is had it not possess such great implications for number theory

[safe merlin]

Yes that's correct but one of the way of stating it is through a limit which can be taught to any undergrad or even a good highschool student who has a good grip on calculus.

[glad adder]

I suppose that's true

But i guess from that perspective, it's an alternative way of representing an open problem, while in this case, it's representing a pretty well-known constant

[safe merlin]

Thank you both of you. I really appreciate you guys took your time to Write a reply.

[glad adder]

Though if you were to show that this limit has implications about the meaning of the E-M constant then maybe that'd be sufficient for research

[safe merlin]

Yes if we can prove that this limit doesn't converge

Then it would imply the irrationality of E-M.

As this Limit is derived assuming that E-M is rational.

So it's irrational if and only of this limit doesn't converge.

[glad adder]

Well I more meant something new about it

Hmm

[safe merlin]

E-M isn't proved to be irrational.

It's an open problem

[glad adder]

Oh I thought it was

In that case go for it

[safe merlin]

Thank you I'll do so

Is there a way I can let someone proof check this?

[glad adder]

I'm sure there's professors that would gladly peer review the work

[safe merlin]

I'm not a university student so don't have a connection with the professors

[glad adder]

Ah I see

That'll be tough then

It's pretty difficult to do research outside of academia

[safe merlin]

It really is

[glad adder]

You also need to be careful about not talking to people that will steal your work. It's uncommon but it happens

[safe merlin]

Yeah that is true gotta be careful about that

[glad adder]

I just think it'd be much more likely for that to occur if someone weren't in academia

[safe merlin]

I guess just publishing it on axriv then posting it on the stack exchange would be the way.

[glad adder]

Yeah someone would post it on stack exchange

I mean if u want you could even post the proof here

Sadly though, id say your work needs peer reviewing by professionals without a doubt

Especially for such an important open problem

[safe merlin]

Yes I don't wanna embarrass myself

When there can be a +- issue in the proof

Lol

I can show you a simple new formula for E-M

[glad adder]

Sure

[safe merlin]

This is a very simple one and nobody had mentioned this.

Its log base 2

Also it diverges for any log base greater than 2

[glad adder]

I see

[safe merlin]

But I don't think there's any use for this

This doesn't converge very fast

[safe merlin]

Three digits of accuracy for 10000 terms

💀

[glad adder]

Yeah thats rough

[safe merlin]

It's interesting though I don't know of any formula in the form of a series in which add and subtract infinity infinite amount of times and still be left with a constant.

Well I guess you can construct one

[median prism]

This is a well known thing. This follows from the first terms of the harmonic sum asymptotic expansion.

[safe merlin]

Is it? I didn't see it anywhere.

[median prism]

it's in wikipedia https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)

It's this formula. Just take into account that (-1)^k/k sums to -ln(2)

[safe merlin]

Ohhh rightttt

[safe merlin]

We can then just write it as n*ln(2)

Loll I didn't have a proof of this thank you

I remember writing and down then working it out it on Wolfram alpha.

[safe merlin]

What do you think about the original limit?

[safe merlin]

Still I haven't seen this anywhere don't know why it's a really simple and unique one.

[median prism]

what do you mean 'a period'? Is it integer?

[safe merlin]

Like 0.234234234 has a period of 3

[safe merlin]

Yes

[median prism]

Whatever integer that alpha is, it is clear that this formula cannot say anything about irrationality of EM constant. Your expression just equals -{10^(na)*ln(10^a)} where {...} is a fractional part. And we dont use here EM constant at all.

How can this say anything about EM constant if we use only some fixed number of digits from that constant?

[glad adder]

Also how does an integer have a non-natural period

[median prism]

Maybe I dont quite see everything, but if your statement is correct than we can apply this to any constant. Say pi or e? Just take their periods? To prove irrationality of some number we have to use the definition of that number at some place.

[median prism]

so it is not a period itself, but its length?

[safe merlin]

Yes

It's the length

[safe merlin]

Well that's from the proof of how we derived this.

[median prism]

On the whole, my opinion is, your limit is too far from helping to prove irrationality. If you disagree you can try to apply your approach to some already solved problems, like irrationality of pi, e or sqrt(2).

[safe merlin]

No this limit can't be applied to any other constant but how we derived this limit may as well derive another limit expression for other constants. Like for example the constant beta defined using series from 1 to infinity 1/p_n of the reciprocals of prime - ln(ln(n)). Has a limit like this one.

[median prism]

And see what the differences are

[safe merlin]

The thing is we just have to prove that the digit block of ln(10) from digit 10^n upto 10^n +n is random and that would be equavillannt to proving that E-M is irrational.

[safe merlin]

Yes I will do that it'll help this futher

[safe merlin]

For natural number n

[median prism]

Still I dont see what 10 has to do with EM constant. Can we take 2 or 3 instead? If it is about randomness of some interval in digits, where do we use the specifics of EM? Why cannot we take any other constant and apply the same reasoning?

[safe merlin]

Yes we can use 2 or 3

Or any natural number for that matter

It would just be a much more messy limit

[safe merlin]

It can be stated as ln(k) has random digits in fixed series of intervals and those intervals will be different for every k.

10 just had nice form in base 10 nothing else that is special about it.

[safe merlin]

Well this limit is strictly derived assuming that E-M is rational. There's only one step in the derivation that can be applied to other constants but after that it's completely different.

[glad adder]

There might be some merit in using 2 instead of 10 potentially

Generally easier to show digital patterns in binary

[safe merlin]

True