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.

湾区游

上周底,又跑湾区一趟,为了面试,也为了玩。这次受益匪浅,拿到了比较好的工作,同时也再次得到学习数学的启发。我的一位不善学,疯疯癫癫的,苏犹半朋友,却建议我跟一位正在MIT学数学的美国IMO金牌得主聊天。此人我已被介绍过,通过一位“知名的高才生中间人”,可与此人未说任意有含量的话,从而想这牛人太忙,与我这庸人无话可说。出乎意料,介绍者却说他与牛人沟通频繁,并且将我加入他们俩人的脸书群,后来又加了一位不相信智商,反徐道辉的,女生物博士生,犹裔也。面试之前的晚上,我住在旧金山的宾馆,附近有一家“现代犹太博物馆”,好奇去看了看,楼上关门,楼下没啥好看的。我们在群里所讨论的好多与种族,文化,智商,和学科相关,还记得我曾经对数学尖子开玩笑,美国IMO选手,非亚裔,比犹裔,而他似乎未感到我的玩笑口气,回说他那界有一两位非犹太白人。他还强调自己很美国人,不是那种在以色列待过的。高中时,他选了中文为他的外语,以童话为他爱听的一首中文歌。学术上,给的感觉是专注,单一的纯数学本科生,非常肯定他会走学术道路。他的具体数学兴趣及倾向为组合数学和理论计算机和解析数论。那天晚上,问他知道那几个二次互反律(law of quadratic reciprocity)的证明,他回答一个引用Zolotarev’s lemma的,并且发了个链接。当时,我只知道高斯和(Gauss sum)的那个,而细节已经记不清楚了。我花了一小时细读那个证明,领悟后感觉漂亮无比,引用的工具极其简单。从来没有想到可以将这数论皇后的定理视为,表达为置换的奇偶的积,毕竟Legendre symbol给的是1或-1,与sgn一样,好妙啊。回顾透明Legendre symbol给的基本是在给循环阿贝尔群(\mathbb{Z}/{p\mathbb{Z}})^{\times}的元素奇偶,二次剩余和偶置换在他们对应母群都是指数为2的子群。这又让我想到高斯对于正十七边形可作图的证明也是引用指数2的子群,其由Galois theory对应于度数为2的域扩张。过两天,我又学到了Gauss lemma,就是\left(\frac{a}{p}\right) = (-1)^n, n\{a, 2a, \ldots, \frac{p-1}{2}a\}大于p/2的元素的数量。证明思路很直接,将\{a, 2a, \ldots, \frac{p-1}{2}a\}的大于p/2的元素负掉,可和其他元素从新凑成\{1, 2, \ldots, \frac{p-1}{2}\}. Eisenstein对该定理的证明,我以前知道其存在但没看懂的,引用类似于Gauss lemma的引理,思路及证明策略抓住,这次却清晰了然。二次互反律的美妙我之前无法欣赏,记得对此定理有过一种稀奇古怪,难以思议之感,是没有抓住并且悟觉它的美妙的对称结构。想起在Eisenstein的证明中,画了一个pq的格子,将-1的次数示为格子的下左象限所有的点数,此为(p-1)(q-1)/4的来源。

数论是纯数学比较活的分支,与现在被我看为形式化繁琐枯燥的测度论相反。回想起,我学测度论习题做不出来开始大大怀疑自己数学的能力并且对数学稍失去了兴趣。而数论从某种角度而言有相反的作用。昨天,我从新过了Bertrand’s postulate的源于Erdos的证,对此同样有新的见解,至少感觉是这样。这次很明确它是以好紧限的有易与素数次数相连的\binom{2n}{n}在假设n2n范围无素数情况下,导致未成立的不等式。证明里的关键观察是2n/3n之间的素数不整除,此有大大减小上限的作用。

我有想过回到学校做数学,可是此愿望看来还是非强于我对自己的能力的怀疑加上挣钱的“心里”需要。在所谓的工业界有了一定的见识是大大开阔了眼界,与那些学术界单纯做学问的人恰恰相反。正面看,这给了我更多的实用性思维,眼观,和技能,而反面,这使得我失去,以不可逆转的方式,利于做研究的隔离所排出的干扰。我小时看到人说离开学校再回去很难,现在知道为何人这么说。所以我的回到学校的企图又失败了,下一个工作来了,有新的实用性的挑战和机会。不过,数学我很可能业余还会学学,盼望在此将出前所未有的新视野。

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

昨天,我学会背了两首中文诗,一首是杜甫的《闻官军收河南河北》,另一首是毛泽东的《沁园春·长沙》

聞官軍收河南河北

劍外忽傳收薊北,初聞涕淚滿衣裳。
卻看妻子愁何在,漫卷詩書喜欲狂。
白日放歌須縱酒,青春作伴好還鄉。
即從巴峽穿巫峽,便下襄陽向洛陽。

杜甫这首诗具体哪一点感动了我,此我难以解释,缺乏文学描述所需要的词汇,同时也还未形成任何诗人的口味。加上,我对当时的中国的历史和文化也几乎一无所知。这还是我学会的第一首杜甫的诗。相反,他的对偶李白的诗我会好几首,如《蜀道难》和《将进酒》。

与杜甫的不同,我对毛泽东的《沁园春·长沙》的能容可有更深刻的理解,由于自己对二十年代的中国的政治形势有过一定的阅读。

沁園春‧長沙
獨立寒秋,湘江北去,橘子洲頭。看萬山紅遍,層林盡染;漫江碧透,百舸爭流。鷹擊長空,魚翔淺底,萬類霜天競自由。悵寥廓,問蒼茫大地,誰主沉浮?
攜來百侶曾游,憶往昔崢嶸歲月稠。恰同學少年,風華正茂;書生意氣,揮斥方遒。指點江山,激揚文字,糞土當年萬戶侯。曾記否,到中流擊水,浪遏飛舟!

这首诗的形势明显跟《沁园春·雪》相同,前段绘画祖国的美丽江山,后段启发式的鼓励壮怀激情的爱国主义革命家,开辟新天地,粉碎军阀混乱之黑暗。也可以说,《念奴娇·昆仑》也有接近或类似的形状。同时也发觉到,原来孔庆东出版的那本描述及讽刺韩国的书的名字却引用的这首诗。

这几月,我在学习俄语,网上找到了毛泽东十八首诗的俄文翻译, 此中有《长沙》的。

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…

阅此非太陌生,令余稍欣慰,表示己有进步,语言能力还不差,当然自己在这方面绝对没什么不得了。此俄语翻译,我还传给了我的几位苏联同志看了看。说起翻译,我昨天还把一段中文翻译成了俄文,至之至大学时给了我,在一个风气腐朽,无知无趣,在我另一位朋友形容为“如家具”的本科生漫天遍野的校园上,不少思想丰富及精神隐蔽的一位绝顶聪明又非昏迷于垃圾美国文化的童年来美的俄罗斯同学。我选择自学俄文大大出于本人本性对此语言及其文化发生的兴趣,但同时,他的鼓励及具体帮助也一直有了一定启发性的作用,有人可分享自己心灵所产生的美感,绝对有一定浪推飞舟的作用。

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.