Alexandre Bali

Email : a l e x . b a l i @ s f r . f r
But I must admit, I never actually use it
If you wanna contact me, use either Messenger...
( h t t p : / / w w w . f a c e b o o k . c o m / A l e x H D P l a y i n g )
... or my Discord : @ b e r l i n g o t d e l a i t # 9 7 6 5

 I'm interested in everything that is closely linked to mathematics, in any field of study. Whatever I do, I publish it here. The fields in which I'm more proficient are currently number theory and analysis, but I wish to get into group theory and algebraic topology as soon as possible.

Papers and drafts :
• Fractal upper bound for Eis(R) the number of Eisenstein integers of norm ≤ R (ResearchGate, 25 May 2020)
 In this short paper, we will show a function $L$ defined over the dyadic fractions in the open interval $(0,1)$ such that $$\text{Eis}(R)\le (2\lfloor R\rfloor+1)\left(2\left\lfloor\frac2{\sqrt3}R\right\rfloor+1\right) -4\underset{0< x< 1}{\sum_{x\;{\rm dyadic}}} \left\lfloor L(x)R\right\rfloor\left\lfloor\frac2{\sqrt3}L(x)R\right\rfloor$$ for all $R\ge0$, with $L$ being defined using a fractal of squares filling the space between a square of side length 2 and a circle of radius 1 inscribed within the square. It also allows us to generate infinitely many quadratic polynomials which upper bound $\textrm{Eis}(R)$. It might be helpful to understand the $A_2$ lattice, and similar ideas could probably be applied to other lattices as well.
• Various patterns and anomalous behaviors from the sine function (Scribd, 30 Oct 2018)
 In this paper, we will present a bunch of sets which tend to reveal somewhat strange patterns from the sine function, and we'll specifically study those who don't really fit in any of those.
• Weak evidence for Goldbach's conjecture (Scribd, 20 Oct 2018)
 In this tiny paper, we will show our recent work on Goldbach's conjecture, which is about whether (or not) all the even numbers greater than or equal to 4 can be equated to a sum of two primes. We will try provide a few weak evidence for the conjecture. We'll conclude on a suggestion to prove Goldbach or the weaker conjecture that asks if infinitely many even numbers can be written as a sum of two primes.
• On the eventuality of a smallest countexample of the Collatz conjecture (Scribd, 29 Sep 2018)
 In this paper, we will show a few results which would be implied if there is indeed a minimal counterexample of the Collatz conjecture, and trying to find out what kind of properties such a number would (or wouldn't) have. We'll also conclude with an idea to find a better algorithm, which we unfortunately had a hard time to decipher.
• A draft on the repartition of algebraic numbers (Scribd, 02 Jul 2018)
 A lot of unsolved number theoretic open problems are about the irrationality or the transcendence of a number. This paper will share a few results which are useful to know about the repartition of algebraic numbers. We will specifically look at quadratic roots as quadratics are easier to study than most other polynomials, and because they also contain ℂ\ℝ roots.