On equations with arithmetic functions [closed]

Question

Is this good topic for research:

equations with arithmetic functions, for example equations like $\varphi(n)=\sigma(n)$ or $\varphi(n)+\sigma(n)=d(n)$ ?

If Anyone here have an advise please tell me that.

Answer 1

From the advices of professors that were provided in comments I think that you should communicate with your advisor. The writing of a PHD thesis is very important and serious, I'm not a professor and I wish you the best. On th other hand I add some ideas in this post. In my opinion I advice you that the strategy is to study simple things in mathematics, and as a tactic to teach ideas about your research or readings of articles to other persons.

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None of the ideas that I expose can constitute a thesis (you will always need the help of a professor). These ideas now are public (meanwhile that my post is well-received, I will not delete it) and to exploit these ideas it is under the laws of copyright and if I refer well the Creative Commons license of Stack Exchange network.

About equations involving arithmetic functions I think that an interesting function is the Dedekind psi function $\psi(n)$ see Wikipedia Dedekind psi function and from here try to deduce insteresting equations involving this arithmetic functions and other arithmetic functions. For example recently I've studied equations as $\sigma(\square)=\text{prime}$ and $\varphi(\text{prime})=\square$, in my opinion it should be interesting to study other kind of identities as $$\psi(A)=\frac{1}{\varphi(A)}\left(2(p+1)\right)^2\left(\left(\frac{p+1}{2}\right)^2-1\right)\tag{1}$$ that have the form $\psi(A)\varphi(A)=\square\times(\square-1)$ (therefore here $A$ is a rectangle) for $A=p\cdot(p+2)$, that is in essence the equation by Tomasz Buchert from Theorem 18 in page 23 of [2]. I tried to get characterization of twin primes by using an associated arithmetic billiard, finally for $p=3$ and thus $p+2=5$ I glue the left and right sides of a rectangle of height $8$ and basis $\psi(A)\varphi(A)=64\cdot 3$ in the shape of a finite cilinder where $\varphi(p\cdot(p+2))$ denotes the number of bounces in arithmetic billiard (if you imagine a small drum crossed by 8 diagonals, if my post is well received I can to scan images and illustrate this post in next days). My belief is that work can be done for other primes constellations (I got it for several sequences in number theory that I can to share in comments). Other idea is try to define equations with more than a variable as I did below my answer of the post on MathOverflow with identifier 119611, as you see in my comments that I've edited concerning equations similar than $$\psi(2(\psi(xy)-(x+y-1))-1)=xy\tag{2}$$ in which are involved two variables $x$ and $y$. Ancient greeks believed in equations involving integers with up to three variables. As examples of equations involving three variables in $\mathbb{Z}$ I add the following proposition and conjecture for even perfect numbers that have the form $Y=\text{prime}\cdot \square$, I add here the article Perfect number from Wikipedia (one can to study these kind of statements for odd perfect numbers having this form $\text{prime}\cdot \square$ with $\text{gcd}(\text{prime},\square)=1$, and for intergers of the form $K^K\cdot \{\text{prime of the form }K^K+1\}$, see A121270 from OEIS or [4]).

Proposition. Even perfect numbers $Y=X\cdot Z$ with $Z$ their associated Mersenne prime and $X$ the square $X=2^{p-1}$ satisfy the equations $\psi(XZ+YZ)=\psi(XZ)+\psi(YZ)$ and $\psi(2XY-XZ)=\psi(2XY)-\psi(XZ)$.

Conjecture. Let $1\leq X,Y,Z$ be positive integers such that $Z$ is an odd integer, $X$ has the form $X=\frac{Z+1}{2}$, $X\mid Y$ and $Z\mid Y$. A) If the identity $\psi(XZ+YZ)=6\cdot\left(\frac{X\cdot\psi(YZ)}{Z\cdot\psi(XZ)}\right)^3$ holds, then $Y$ is an even perfect number. B) If the identity $\frac{\psi(2Y)-\psi(Y)}{\psi(XZ)}=\frac{\sigma(X)\sigma(Z)}{2Y}$ holds, then $Y$ is an even perfect number.

Other idea is to state and study problems about primality or divisibility involving arithmetic functions, as Dedekind psi function (I can to clarify this adding literature in comments).

In the section Comments of OEIS for the sequence with identifier A001615 is added the meaning in terms of lattices for the Dedekind psi function. I am trying to attach it to arithmetic billiards, in particular I try to interpret it in terms of the arithmetic billiards that I defined in the post on MathOverflow with identifier 429420 and title Arithmetic billiards, prime numbers and the Goldbach conjecture, see [3]. I'm trying to attach more ideas to these diagrams, for example defining refractions (I can to illustrate this idea; other idea is to define from these refractions faces of tetrahedra and cross products of vectors), I did an attempt to think in the definition similar than $\psi_n(x,t)=\sum_{k=1}^{\varphi(n)} e^{i E_k \cdot t}e^{x^k}$ and the operator $L(\psi_n(x,t)):=\psi_n(x,0)= f_n(x)$, I want that these $f_n(x)$ are the functions defined in THEOREM from [1], where here $\varphi(n)$ denotes the Euler's totient function, $i=\sqrt{-1}$ the imaginary unit and the "energy" $E_k$ are constant that I don't know how to define (I did some atempts to define these). Ancient Greeks quantized space and time lapses, but did not know the stationary-action principle/the Principle of Least Action beause they did not quantify the action (I will edit the final version of this parenthesis in the next six months: at my home I'm trying to think in Fermat's principle and Snell's law; the Gudermannian function; the Lambert W-function; the Apéry's constant and how did Euler get $\zeta(2)$ from $\frac{\sin(x)}{x}$ in an attempt to made more interesting this kind of billiards).

References:

[1] David M. Bradley, A Curious Way to Test for Primes Explained, Mathematics Magazine Vol. 82, No. 3, June 2009.

[2] Tomasz Buchert, On the twin prime conjecture, Master's thesis, Poznan 2011 (if I refer well the author published it in his website).

[3] Post 429420 with title Arithmetic billiards, prime numbers and the Goldbach conjecture asked a month ago in MathOverflow.

[4] Michal Krizek, Florian Luca, and Lawrence Somer, 17 Lectures on Fermat Numbers, CMS Books in Mathematics, Springer (2001), page 156.