## A derivation of a Riemann zeta function identity

Yesterday, I saw the following Riemann zeta function identity:

$\displaystyle\sum_{n=1}^{\infty} \frac{\sigma_a(n)\sigma_b(n)}{n^s} = \frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}$.

I took some time to try to derive it myself and to my great pleasure, I succeeded.

Eventually, I realized that it suffices to show that

$\{(dd_a, dd_b, d^2 n) : d_a | n, d_b | n : d, d_a, d_b, n \in \mathbb{Z}\}$

and

$\{(dd_a, dd_b, n) : dd_a d_b | n : d, d_a, d_b, n \in \mathbb{Z}\}$

are equal as multisets. As sets, they are both representations of the set of $3$-tuples of positive integers such that the third is a multiple of the least common multiple of the first two. In the latter one, the frequency of $(a,b,c)$ is the number of $d$ that divides both $a$ and $b$ such that $ab | cd$. In the other one, if we write $(a,b,c)$ as $(d_1 d_2 a', d_1 d_2 b', c)$ where $\mathrm{gcd}(a', b') = 1$, the $ab | cd$ condition equates to $d_1^2 d_2 a'b' | c$, which corresponds to the number of $d_1$ dividing $a$ and $b$ and such that $d_1^2 | c$ and with that, $d_2a', d_2b'$ both dividing $d_1^2 / c$, which is the frequency of $(a,b,c)$ via the former representation.

The coefficients $\{a_n\}$ of the Dirichlet series of the LHS of that identity can be decomposed as follows:

$a_n = \displaystyle\sum_{d^2 | n, d_a | \frac{n}{d^2}, d_b | \frac{n}{d^2}} (dd_a)^a (dd_b)^b$.

The coefficients $\{b_n\}$ of the Dirichlet series of the RHS of that identity are

$b_n = \displaystyle\sum_{dd_a d_b | n} (dd_a)^a (dd_b)^b$.

Observe how both are equivalent in that via the multiset equivalence proved above, $n$ determines the same multiset of $(dd_a, dd_b)$ for both and across that, the values of the same function $(dd_a)^a (dd_b)^b$ are summed. Hence the two series are equal.

## Automorphisms of quaternion group

I learned this morning from Brian Bi that the automorphism group of the quaternion group is in fact $S_4$. Why? The quaternion group is generated by any two of $i,j,k$ all of which have order $4$. $\pm i, \pm j, \pm k$ correspond to the six faces of a cube. Remember that the symmetries orientation preserving of cube form $S_4$ with the objects permuted the space diagonals. Now what do the space diagonals correspond to? Triplet bases $(i,j,k), (-i,j,-k), (j,i,-k), (-j,i,k)$, which correspond to four different corners of the cube, no two of which are joined by a space diagonal. We send both our generators $i,j$ to two of $\pm i, \pm j, \pm k$; there are $6\cdot 4 = 24$ choices. There are by the same logic $24$ triplets $(x,y,z)$ of quaternions such that $xy = z$. We define an equivalence relation with $(x,y,z) \sim (-x,-y,z)$ and $(x,y,z) \sim (y,z,x) \sim (z,x,y)$ that is such that if two elements are in the same equivalence class, then results of the application of any automorphism on those two elements will be as well. Furthermore, no two classes are mapped to the same class. Combined, this shows that every automorphism is a bijection on the equivalence classes.

## A recurrence relation

I noticed that

$(x_1 - x_k)\displaystyle\sum_{i_1+\cdots+i_k=n} x_1^{i_1}\cdots x_k^{i_k} = \displaystyle\sum_{i_1+\cdots+i_{k-1}=n+1} x_1^{i_1}\cdots x_{k-1}^{i_{k-1}} - \displaystyle\sum_{i_2+\cdots+i_k=n+1} x_2^{i_2}\cdots x_k^{i_k}.$

In the difference on the RHS, it is apparent that terms without $x_1$ or $x_k$ will vanish. Thus, all the negative terms which are not cancelled out have a $x_k$ and all such positive terms have a $x_1$. Combinatorially, all terms of degree $n+1$ with $x_k$ can be generated by multiplying $x_k$ on all terms of degree $n$. Analogous holds for the positive terms. The terms with only $x_1$ and $x_k$ are cancelled out with the exception of the $x_1^{n+1} - x_k^{n+1}$ that remains.

This recurrence appears in calculation of the determinant of the Vandermonde matrix.

## 两首诗

### Чанша

В день осенний, холодный
Я стою над рекой многоводной,
Над текущим на север Сянцзяном.
Вижу горы и рощи в наряде багряном,
Изумрудные воды прозрачной реки,
По которой рыбачьи снуют челноки.
Вижу: сокол взмывает стрелой к небосводу,
Рыба в мелкой воде промелькнула, как тень.
Всё живое стремится сейчас на свободу
В этот ясный, подёрнутый инеем день.
Увидав многоцветный простор пред собою,
Что теряется где-то во мгле,
Задаёшься вопросом: кто правит судьбою
Всех живых на бескрайной земле?
Мне припомнились дни отдалённой весны,
Те друзья, с кем учился я в школе…
Все мы были в то время бодры и сильны
И мечтали о будущей воле.
По-студенчески, с жаром мы споры вели
О вселенной, о судьбах родимой земли
И стихами во время досуга
Вдохновляли на подвиг друг друга.
В откровенных беседах своих молодёжь
Не щадила тогдашних надменных вельмож.
Наши лодки неслись всем ветрам вопреки,
Но в пути задержали нас волны реки…

## Follow-up on the chosen people

Readers of my blog might recall my post titled The Chosen People. I’ve had the privilege to have had some fruitful and stimulating discussions with a friend of mine who was raised Christian on this matter. Such reminds me of Bobby Fischer, and I had just found my way to this, which has a transcript of some of Fischer’s antisemitic remarks. I was also reminded of sayings of a CCTV reporter along the same lines. When I was a child, I enjoyed watching the kids show Arthur. Particularly memorable was this rich girl Muffy Crosswire. I remember particularly vividly this clip, where when that girl’s psychopath auto CEO daddy turned on the speaker that exposed the dealings within the “strategy room,” it was like: so what if the engine falls out, once they’re off the lot, it’s their problem! Hmm, maybe that’s why American cars couldn’t compete with Japanese ones, because there were such people in executive management at Ford and General Motors? Also, there was the special Christmas episode of Authur, from which I learned that Muffy is the Gentile while her bestie Francine, the poor daughter of garbage man, is the Jew. Given how so a high percentage of the Russian oligarch robber barons are Jewish as are so many of the financiers and media moguls in America, aren’t the roles a bit reversed? Now, given that, would Marc Brown have had difficulties in his career had the portrayal been more in phase with the statistical reality? On this group that is the subject matter of the blog post, I have also heard people say in private things in the likes of: it’s always they win others lose, that’s why nobody likes them. And they’re basically doing to the Palestinians what the Nazis did to them!

The Jews are a very unique group. They seem to take almost exclusively jobs in the intellectual class. You see very few in the low end menial jobs. Now that is a natural consequence of their high average IQ over 115ish. This naturally breeds resentment. People love to say how now Asians are the new Jews, but keep in mind there most Chinese do menial labor for living, especially in China. Even in the US, many of them work at restaurants and laundromats and such and they are very socially disjoint with the Chinese intellectual class, the ones who work for Microsoft or profess at universities. Also, there are almost no Chinese in positions of power in the US right now. We all know that ruling classes of virtually all cultures are scumbags in one way or another. However, I would say that some are more benign than others. Remember that during the Cultural Revolution in China, children of the communist elite were actually forced to go work in the countryside to get a taste of what the majority of working class people go through on a daily basis, and many of the current leaders of China are in that category. Steve Hsu has pointed out a possible lower prevalence of psychopaths among those of East Asian descent as a partial explanation to the dearth of East Asians in “leadership” positions in America. There is also the (very politically incorrect theory) that the Holocaust wiped out the weaker and nicer portion of the Jewish population that further contributes to the phenomenon here described. Now we all know that people who are too smart for their own good are a root of resentment but those people are quite benign and contribute much to our culture and are responsible for the radical scientific and technological advances which enrich (hopefully) people’s lives. However, there is more cause for hate when you have a group of parasites who do no menial labor and hoard the wealth created by the blood and sweat of working masses for themselves and even feel entitled to do so.

I have not much more to say at this moment on this matter. You, the reader, be the judge.

## Humanities

I’ve been very distracted by humanities lately and I am even contemplating making a career in it, which would have been farthest from my mind in high school or college. In high school, I struggled greatly with English class when in the novels I was forced to read, I couldn’t understand what was going on half of the time, due to misunderstanding of definitions of words or lack of requisite familiarity with the cultural context. There is also that I only started learning English at age 6, without being exposed to it much at home, which means I don’t know the English names of certain household items. Literature made little sense to me, but history I rather liked, though I disliked the American rendition of history.

I started reading Chinese online in high school. It was very difficult for me at that time, but gradually I was able to make sense of things. For language at school, I took Spanish and I was horrendous at it. I’ve forgotten most of it by now, but I could easily relearn it with my much enhanced level of verbal maturity. I’m not very motivated to learn Spanish anymore because there isn’t very much high culture in it (pardon my snobbery).

I hardly took humanities courses in college, but I did take a few that were required. I remember vividly how in the writing class I took most of the students couldn’t write coherently at all. I did starting in my third year of college develop an interest in Chinese poetry. I remember spending quite a while to memorize 蜀道难, without understanding what it was really saying as it was full of type of Chinese language literary that I had no exposure to at that time. I started reading Baidu Baike without feeling like it was too overwhelming.

In high school and even in college, it never occurred to me to take humanities seriously. There are no jobs in that. And students who major in it, at least not in the very top schools, are pretty fucking dumb. Additionally, I always considered language my weak point. In high school, I remember calculus being super easy for me while English class being almost torture. I had to really force myself to write those literary analyses and memorize the details of what happened in the each chapter of Dubliners by James Joyce so that I could actually ace the quizzes. Honestly, how do you fucking expect high school students to really understand what’s going on in that.

Early in my fourth year of undergraduate, I started learning Russian out of a desire to understand the lyrics of some beautiful Russian songs. Also, the grammar of the language, being highly inflectional, was quite fascinating. Also, the Soviet Union, which shaped the course of 20th century history so substantially, was viewed by me as an entity to be understood to a reasonable level.

I know American education really emphasizes liberal education where in undergraduate you study very generally. It is in contrast to many other systems, where students focus mostly on their major in college. In China, the education system of which was influenced by the Soviet Union in the 50s, has very specific majors, where say you major and engineering and by the time you graduate you are already sort of an expert in a very narrow part of engineering, such as automobile engines. So because of that, I didn’t actually learn a ton of math or computer science despite majoring in them, although it also had to do with my having been quite dysfunctional back then.

On humanities, I’ve also seen people who major in them as rather spoiled. “English literature” was cited by a Chinese immigrant engineer I worked with as something for rich people who don’t have to worry about making a living. Yes, there are actually parents who pay for their kids’ tuition at a non-state school which is like $50k / year now for them to study a useless subject with no employment prospects, and I’ve even seen cases of parents paying them for continual study. To be fair, just being able to study is already privilege. After all, most people in this world not by choice are engaged in menial labor that is often also dangerous and bad for health. It’s especially egregious when these useless humanities students (many of whom aren’t even any good at humanities or language) also support very unprogressive and even vile politics. There are probably many of those at many of America’s Ivies (I’ve been told that at Yale the major for kids who party too much to graduate is American Studies). On that other hand, there are obvious advantages of entitlement, which I won’t go into. I grew up in very ordinary middle class American neighborhoods, surrounded by very ordinary people. I wasn’t like the other kids at all, though the environment made me fit in artificially. That there are parents who spend over$50k / year on enrichment for their (often idiot) children was so foreign to me at that time. I believed firmly in, by virtue of upbringing, meritocracy, and the gradual realization of the world’s so not being that way was rather disillusioning. Why play it fair when you don’t have to? Instead of feeling of disgust at such, I have evolved to view it as very normal and inherent to human nature.

As for humanities, I haven’t been reading novels in their entirety, which I don’t really have the patience for, but I have learned countless words in Chinese, English, and Russian, which I have a much better memory for now. The humanities culture in China, a rich civilization and culture of millennia, which I have indirect access to via my internet connection, has deeply impressed me. I read classic poetry in it and imagine how anybody could have possibly written this. It is evident that I am still quite a ways away from the big brains in anything, and it is possible that I am simply not naturally talented enough. I will say though that most kids in my current generation, especially in America, are as dumb and ignorant as rocks. Is this due to genetic deterioration or due to the degenerate culture that pervades my generation? I feel like much is the inability of many to recognize that latter and the mental courage to counter it at the expense of some degree of social isolation. We are in an era of civilizational decline, and when I say this, I echo the private exchanges among many highly gifted and aesthetically discerning people who must hide to some degree in the suffocating atmosphere of this country, especially in the corporate world.

Most of my friends are in STEM, in fields like math or software engineering, so the types within my milieu constitute a rather lopsided group. I have little contact with any in the world of humanities and social science in America and I am curious as to what kind of people they are. Well, Steve Hsu says many of them are high V lower M types, confused, in the likes of Stephen Gould. My temporary loss of interest in mathy material is worrisome. Am I degenerating into those I looked down on as saying nothing of substance. (In case you haven’t noticed, I am saying absolutely nothing of substance right now, but perhaps it needs to be said for social and political reasons, as part of an cultural/ideological war in some sense?) Perhaps this is an artifact towards my intelligence shifting more towards the verbal end of things? Could it possibly be that now with some more years of organic cognitive maturation I am now metamorphosing towards the reverse direction of lopsidedness?

When I talk with people in the software profession, it is apparent that they are types for whom learning natural language is unintuitive, despite programming being a very languagy pursuit, with many programmers conspicuously bad at math, as the work is of a very qualitative nature. There had been to me many parallels to writing in software engineering to my blind intuition but such is clearly not the case empirically.

There are many people, especially certain parents, who will say that once you’re an age, you can’t just study and that you have to go out and earn money and do something that’s directly useful to others, that makes some tangible contribution to society. That can be writing software or building houses or scrubbing toilets. But it can’t be hiding in academia, especially in a subject like pure math or comparative literature. There are all these stereotypes of about those in pursuit of or already in possession of Permanent Head Damage as kids who don’t want to grow up many of whom also lack common sense or have no practical skills, who manage to put spoons and forks flat on the dishwasher. There are also stereotypes in virtually all societies that those people only know books and are often too uppity to engage in forms of labor regarded as more menial.

I thought that maybe I could go into history or political science or language. Maybe I could even make a career in that. It’s not impossible. Or maybe even go into the legal profession?

I’m at a loss on where to take my life as software, which I’m not bad at, does seem not to marry my personality. Most software engineers will see me as too weird for say learning Russian on my own. Now am I one of those dysfunctional weirdos who can only survive in academia, which we already know has no jobs? Also, at this point, I am mostly a consumer, rather than a producer, of knowledge, a state of intellectual incapacity that will hopefully alter itself. There have been times when I could not help but see myself as good for nothing, though surely my standards on that are quite high. Hopefully, I can find something in which I can truly excel and even make a name for myself. Let’s see if that happens.