I hope I am doing something right... just dont want my draft to be stolen by anyone but still risking it to at least get some feedback on how I could perhaps make the draft a bit more consistent. And yes I did use AI for it but again all the work, ideas and theorems were made and found by me and no one else. The AI was only used to make the text and to ensure a good comprehension of the idea at hand. And no for the record this text itself isnt AI
. I just want like I said honest feedback with some friendly discussion about my framework here so I can see others point of view. And like I said (for the second time because I am dead serious) please dont steal my work
🙏 it took 6 months to build it from scratch (since I am autodidact.)
#Request review for a draft
kind vigil
kind vigil
I forgot to put context on what the draft actually is so let me give a short description here:
It develops an operator version of the geometric series that lets you normalize infinite products (like the sine and zeta products) and then differentiate them to get new Dirichlet/Lambert‑type series with explicit closed forms.
dim ermine
What are you doing?
kind vigil
What are you doing?
exploring infinite geometrical products and their renormalized relationships with the original operator. It allows me to link both sides to obtain closed froms for infintie products of the form $\prod_{k\geq 1} \frac{a^{g(k;x)} - 1}{a-1}$ and to also obtain closed form for specific types of modified Dirichlet series or just in other words Lambert series when applying the derivative with respect to x. The framework only appeared when studying the relationship between a telescoping geometrical product and a geometrical function with a product as its upper bound. Hope that makes sense
charred torrentBOT
twitch Tamplr
kind vigil
unless you meant something else with this question...
Anyways, more specifically after one expands the inverse and non inverse infinite product (which both yields the exact closed form according to the framework established) one could indeed find the following claim that is after noticing that $T_{f}^{\prod_{k\ge1}^{ }g\left(k;x\right)}=\prod_{k\ge1}^{ }T_{f^{C\left(x\right)}}^{g\left(k;x\right)}=\prod_{k\ge1}^{ }T_{f^{C\left(x\right)}}^{1-\left(1-g\left(k;x\right)\right)}=\prod_{k\ge1}^{ }\left(1+f^{C\left(x\right)}T_{f^{C\left(x\right)}}^{g\left(k;x\right)-1}\right)=1+f^{C\left(x\right)}\sum_{k\ge1}^{ }T_{f^{C\left(x\right)}}^{g\left(k;x\right)-1}++f^{2C\left(x\right)}\sum_{k_{1}>k_{2}}^{ }T_{f^{C\left(x\right)}}^{g\left(k_{1};x\right)-1}T_{f^{C\left(x\right)}}^{g\left(k_{2};x\right)-1}+...$
charred torrentBOT
twitch Tamplr
kind vigil
then it would imply the following relationship
$\left(\sum_{n\ge0}^{ }\frac{f^{nC\left(x\right)}}{\left(f^{C\left(x\right)}-1\right)^{n}}\sum_{k_{1}<...<k_{n}}^{ }\prod_{m=1}^{n}\left(f^{\left(g\left(k;x\right)-1\right)C\left(x\right)}-1\right)\right)\left(\sum_{n\ge0}^{ }\frac{f^{nQ\left(x\right)}}{\left(f^{Q\left(x\right)}-1\right)^{n}}\sum_{k_{1}<...<k_{n}}^{ }\prod_{m=1}^{n}\left(f^{\left(\frac{1}{g\left(k;x\right)}-1\right)Q\left(x\right)}-1\right)\right)=1$
charred torrentBOT
twitch Tamplr
kind vigil
Assuming for n = 0 both sides would yield 1 as a given first term
rip latex too big 
also one could notice that the first term is indeed somewhat a modified theta function but only of the form of an exponential minus one which is why "modified" is used
$\frac{f^{C\left(x\right)}}{f^{C\left(x\right)}-1}\sum_{k\ge1}^{ }\left(f^{\left(g\left(k;x\right)-1\right)C\left(x\right)}-1\right)$
charred torrentBOT
twitch Tamplr
kind vigil
did you at least read the draft?
kind vigil
nothing is defined at all
what tells you that?
kind vigil
god awful notation
I mean... I prefer it that way ngl but you can change it if you want
again, I am only here for feedback
so rn all that you are telling me is that the draft is unorganized or perhaps not clear enough
I guess I could just go ahead and make some changes or maybe just start a better cleaner draft maybe
dim ermine
what tells you that?
what do any of the variables go to?
Maybe it's my lack of knowledge, but it seems unclear.
kind vigil
name some of the variables you probably are refering to
dim ermine
C(x), g(x), Q(x), T, really everything
kind vigil
Roughly:
-
$T_f^x = \dfrac{f^x - 1}{f - 1}$ is my "geometric operator" – it is just the finite geometric series in closed form, extended to complex $x$. I use it as the basic building block for all products.
-
$g(k;x)$ (or also just $g(k)$ for short (since I write my stuff on a whiteboard)) is the $k$-th factor in an infinite product. For example, for $\sin x/x$ I took
$
g(k;x) = 1 - \frac{x^2}{(k\pi)^2},
$
so that
$
\prod_{k\ge 1} g(k;x) = \frac{\sin x}{x}.
$
The normalization theorems are stated for such $g(k;x)$. -
$C(x)$ is the forward normalization constant. From the partial products
$
G_n(x) := \prod_{k=1}^n g(k;x), \qquad
S_n(x) := \sum_{k=1}^n \bigl(g(k;x)-1\bigr),
$
I set
$
C_n(x) := \frac{G_n(x)-1}{S_n(x)}, \qquad C(x) := \lim_{n\to\infty} C_n(x).
$
It is the unique constant exponent that makes the normalized product
$\prod_k T_f^{g(k;x)}$ match $T_f^{G(x)}$ to first order at $f=1$. -
$Q(x)$ is the same idea but for the reciprocals $1/g(k;x)$; it is the effective exponent for the inverse product, used in the inverse normalization and the symmetric identities.
And for the record, if you really hate $f$ as a base, you can just change the variable $f$ if you want it wont make a difference (that is if you keep the restrictions we imposed on it).
charred torrentBOT
twitch Tamplr
kind vigil
make sense so far or is it still confusing? Let me know
but if you keep reading, they should appear and it should tell you exactly what they are and what they are used for
kind vigil
Maybe it's my lack of knowledge, but it seems unclear.
no you're fine. Originally the draft starts with all the definitions THEN it shows WHY they are here and HOW they are here so it makes sense to me that you asked those questions to be honest. No pressure
dim ermine
Yeah I'll keep reading later
kind vigil
its been a while now ggs
dim ermine
Yeah
kind vigil
I guess I need to rework on it and make the notations rigorous because the method and the derivations are quite interesting itself but its just a matter of how I presentate it or how I show it that would make the difference
sad
misty adder
- is simply false, its unclear what you are evne trying to express, constantly swtiching between approx and equal signal but the exact statements you make only hold in first orders
everyhting downstream is false as a result and everything prior is basic algebra
atleast it doesnt strike me as utter slop though
kind vigil
atleast it doesnt strike me as utter slop though
fair enough. FYI this now became another draft in my collection so it serves as just another source I can collect a bit of what I have written before. And I am not surprised it contains many errors and poor claims up to 3. If your interested I made an even better manuscript with full proof, lemmas, appendix, chapters, etc but its still a draft non of the less so I wouldnt be surprised if you spotted mistakes.