# Using linear algebra to study number theory?

I've posted a paper on arXiv that outlines a linear algebra approach to number theory.

Specifically, I have the following questions:

1. Is it possible to draw connections between the factorization matrix (Def. 4 and 6) and the Redheffer matrix or the Mertens function? Does this allow to come up with an alternative formulation of the Riemann hypothesis (Eq. 55 and 56)?

2. Is it possible to evaluate the difference $\pi(x) - \mathrm{Li}(x)$ using the matrix algebra expression for $\pi(x)$ (Eq. 44)? How can the inverse of the factorization matrix (Eq. 39) be further developped to help in this respect?

Answers to either one of the questions are appreciated!

• Is this just a promotion for the arXiv publication? Seriously, starting with "just a neuroscientist with an interest in statistics" and then jumping to the Riemann hypothesis might for many make the paper appear not worth reading ... – Hagen von Eitzen Sep 19 '17 at 10:04
• I wouldn't call it "promotion" as there's no possibility for me to earn something with it. Would you suggest to remove the neuroscience/statistics reference in order not to distract from the subject matter? If you believe that linear algebra can help to study arithmetic functions, it's worth a read. – Joram Soch Sep 19 '17 at 10:33
• @HagenvonEitzen I hope that you don't find rude this my comment, but I don't agree with your critic about the profession of the user asking here. If I understand well the user tell us the profession because need help or a different point of view about the work. Of course critics help to see the things from different view point, but the point in site is if you can help with an answer, in this case about the approach, the mathematics, of the paper. Good day. – user243301 Sep 26 '17 at 19:14
• @JoramSoch What Markus meant was that you should include equations and and definitions that you refer from the paper in the question. – A---B Sep 28 '17 at 13:31
• @user243301 I don't find your comment rude and I didn't want to criticize the OP for his profession. – Hagen von Eitzen Sep 28 '17 at 13:33

Regardless of what others have written, I feel Joram Soch is (intuitively) on the right track.

I would like to comment on his specific question:

1. Is it possible to draw connections between the factorization matrix (Def. 4 and 6) and the Redheffer matrix or the Mertens function? Does this allow to come up with an alternative formulation of the Riemann hypothesis (Eq. 55 and 56)?

My suggested answer: I am not very sure if one can discover a connection or a pattern in the “Factorization Matrix”, because the factorizations of a number n, while going from one given number n to a number n+1, then n+2 etc. seem to be unpredictable. At least the sequence of Liouville functions lambda(n) appears like tosses of coins. Where lambda(n) = +1, if n has even number of prime factors and lambda(n) =-1 if n has odd number of factors. See paper in Arxiv: [https://arxiv.org/pdf/1609.06971v9.pdf][1] . However, Joram Soch is certainly very right in seeing a connection between the above problem and the Riemann Hypothesis, because a careful analysis of the properties of numbers and their factorization leads a path towards a resolution of the Riemann Hypothesis. See my “Road Map”: [https://www.researchgate.net/publication/318283850_A_Road_Map_of_the_Paper_on_Coin_Tosses_and_the_Proof_of_the_of_the_Riemann_Hypothesis][2]

K. Eswaran

• The Mertens function is essentially defined by a linear recursion $\sum_{k \le n}M(n/k) = 1$. The matrix formulations of the RH are useful to treat $M(n), n \le N$ all at once and to see what happens when perturbating those (Dirichlet series for which the RH fails). Those matrices defined in term of $1_{i | j}$ and $\mu(n)$ and the Mertens function have interesting norm, eigenvalues.. Note the goal is to show $(\mu(n))_n$ is almost orthogonal to the constant sequence. Also the matrix $D_{i,j} = 1_{i| j}$ has some sort of fractal structure. – reuns Sep 28 '17 at 16:11
• I was more concerned with the Riemann Hypothesis and how the behavior of the Liouville function $\lambda(n)$ determines the location of zeros of the zeta function; and not so much with the Mertens function. – K. Eswaran Sep 29 '17 at 2:01
• $\lambda(n)$ is essentially a more complicated version of $\mu(n)$ and its only advantage is being completely multiplicative – reuns Sep 29 '17 at 14:18
• However, but one advantage that $\lambda(n)$ has is the fact that since it takes only the values +1 or -1 (and not 0), it is easier to compare a consecutive sequence of $\lambda(n)'$s with coin tosses. This in addition to it's other properties, makes the application of Littlewood's theorem much easier. For details please read the paper cited above. – K. Eswaran Sep 29 '17 at 17:46