# 난제로의 초대 제 3강의 pdf

난제로의 초대의 John Lee입니다.

학생분의 요청으로 목요일에 있을 소수 정리(PNT) pdf 본을 미리 업로드하겠습니다.

20세기 수론의 최고의 발견이라 불리는 디리클레 등차수열 정리(DTAP)와 소수 정리(PNT)를 한데 묶어, 증명 과정 및 지금까지의 발견들을 올렸습니다. 개인적으로 가장 관심이 많은 분야이기도 해서 증명을 조금 세세히 실었습니다.

모두들 그럼 7월 7일: 소수 정리 편에서 뵙도록 하겠습니다.

난제로의 초대 제 3 강 pdf

∴이 교본의 무단 상업적 배포나 편집에 있어서는 법적 조치를 취하도록 하겠습니다.

∴∴해당 교본은 완전한 검토가 진행된 것은 아니어서 조금의 오류와 오타를 포함하고 있을 수 있습니다. 오류와 오타를 발견하셨다면 답글로 남겨주시길 바랍니다. 최대한 빠른 시간 내에 수정하도록 하겠습니다.

# 난제로의 초대 제 1, 2강의 pdf

안녕하세요, 난제로의 초대의 John Lee 입니다.

부족한 강사의 정신없던 첫 강연이었지만, 중간에 아무도 자리를 박차고 나가지 않아주셔서 정말 감사합니다…ㅎㅎ

오늘 수업에서 말씀드렸다시피, 수업에서만 다뤄진 내용도 있고 pdf에서만 다뤄진 내용도 있습니다. 가능하시다면 수업도 참석하시고 pdf도 읽으시는 것을 권해드립니다.

6월 30일인 오늘 난제로의 초대 제 1강인 “페르마의 마지막 정리 상편”이 잘 마무리 되었습니다. 강의 pdf는 제 2강인 하편과 통합하여 업로드합니다.

오늘 참석해주신 여러분들 정말로 감사드리며, 더 나은 모습으로 7월 5일 “페르마의 마지막 정리 하편”에서 뵙겠습니다. 감사합니다.

난제로의 초대 제 1 단원

∴이 교본의 무단 상업적 배포나 편집에 있어서는 법적 조치를 취하도록 하겠습니다.

∴∴해당 교본은 완전한 검토가 진행된 것은 아니어서 조금의 오류와 오타를 포함하고 있을 수 있습니다. 오류와 오타를 발견하셨다면 답글로 남겨주시길 바랍니다. 최대한 빠른 시간 내에 수정하도록 하겠습니다.

# An Invitation to the Problems

안녕하세요, 경희대에서 여름 수학 세미나: 난제로의 초대를 강연하게 된 John Lee라고 합니다.

작년처럼 기회가 닿아 다시 한번 강연을 준비하게 되었습니다. 부족한 실력이지만 모쪼록 잘 부탁드립니다.

Hello, this is John Lee, and I savored an opportunity to conduct another seminar at Kyunghee University. Great pleasure to meet you, and I hope to see you on 6/30.

# The Half-Plane of Convergence of Prime Zeta Function

This post contains a brief proof of Conj. 3 posted on Lee’s Conjecture. To avoid confusion I used $x$ as real variable, and $s = \sigma + it$ as complex.

I proposed a conjecture as follows:

Conj. 3:
Let $\mathbb{P}$ be a set of primes. The supremum of $x$ such that $\displaystyle \sum_{p \in \mathbb{P}} \frac{1}{p^x}$ diverges is $x = 1$.

Let us define a Prime Zeta Function as follows: $\displaystyle P(s) = \sum_{p \in \mathbb{P}} \frac{1}{p^s}$.
In fact, the conjecture turned out to be an easy comparison series problem.

Consider the original Riemann Zeta function, that is defined as $\displaystyle \zeta(s) = \sum_{n \in \mathbb{N}} \frac{1}{n^s}$. It was already proven that $\zeta(s)$ is defined on a half plane of $s$ such that $\sigma > 1$. The domain was extended over all complex plane except $s = 1$ via Analytic Continuation in Riemann’s paper.

Since the conjecture is asking the supremum of $x$, let us only consider the case of real numbers. Obviously, $\zeta(x) \geq P(x)$ for $x > 1$, as each term in the summand of $P(x)$ is also included in $\zeta(x)$, and every term in $\zeta(x)$ is positive. Thus, as long as $\zeta(x)$ is defined, so is $P(x)$. Hence, the domain of $P(x)$ includes a half real line $x > 1$. Therefore, the supremum of $x$ where $P(x)$ diverges must be $x = 1$; the conjecture is true.

While studying Introduction to Analytic Number Theory by T. Apostol, I encountered a following theorem. (1976 Edition, pp.233)

Theorem 11.8:
If [a Dirichlet] series $\sum f(n)n^{-s}$ converges for $s = \sigma_0 + it_0$, then it also converges for all $s$ with $\sigma > \sigma_0$. If it diverges for $s = \sigma_0+it_0$, then it also diverges for all $s$ with $\sigma < \sigma_0$

$f(n)$ presented in the theorem is an arithmetical function. Consider a prime detecting function $\rho (n)$ such that $\rho(n) = 1$ for $n$ being prime, and $\rho(n) = 0$ for $n$ being composite. Then, the Dirichlet series $\sum \rho(n)n^{-s} = P(s).$ Therefore, we may apply Theorem 11.8 to deduce the following fact.

We already showed that for any real number $x > 1, P(x)$ converges. Take arbitrarily small $\epsilon > 0$, then $P(1+\epsilon)$ converges. Thus, by Theorem 11.8, for any $s$ with $\sigma > 1+\epsilon, P(s)$ also converges. Letting $\epsilon \to 0$, we can conclude $P(s)$ is defined for all $\sigma > 1$. Thus, the domain of $P(s)$ is now extended to the half plane $s$ where $\sigma > 1$.

Then, we can come up with a few new problems.

1. Can we extend the domain of $P(s)$ via Analytic Continuation?
2. As we defined Prime Zeta Function, let us define Composite Zeta Function, that is $\displaystyle C(s) = \sum_{n \in \mathbb{N} \setminus \mathbb{P}} \frac{1}{n^s}$. Will the Composite Zeta Function have the same domain?
3. Does Prime Zeta Function have functional equation as Riemann Zeta Function? If so, what are the trivial zeroes and nontrivial zeroes of $P(s)$?

# How I am doing…1

No posts for a long time. This is just a friendly notice that the blog is not “closed.”

Recently, I was dealing with a tough schedule, known as “Graduate Program Admission.” Thus, I did not have enough time to carry on my studies until the early January. Now, I’m mostly done with the application, and resumed studying in Analytic Number Theory, with the aid of “Introduction to Analytic Number Theory,” by T. Apostol. I found this book utmost fascinating for two reasons. Firstly, even though it was written in 1970’s, the style is pretty modern. Unlike many other old textbooks I discovered, or excavated to be more precise, from the library, this textbook has a modernistic sense in displaying equations and proving theorems. Secondly, the author proves every single theorem. I repeat, EVERY SINGLE LEMMA, PROPOSITION, PROPERTY, AND THEOREM. I really recommend this book for the undergraduate students studying Analytic Number Theory for the first time.

Well, besides my academic life, I recently found a new hobby: cooking. Check my “About Lee” post, the hobby “bowling” is changed to “cooking.” What’s my plan is, suggested by a few friends of mine, to create a separate blog, entitled “Mathematician in a kitchen,” or something like that, and post about cookings and recipes.

So, that’s how I am doing, just having another fine day. Thanks for reading, and I will come back with some cool Lee’s Challenge problems once I study quite a bit of Analytic Number Theory.

# 2015 Math Seminar; Good Day To Math Syllabus

Here is a complete version of syllabus of “Good Day To Math”: Good_Day_To_Math_Sylb

# List of Lee’s Conjectures

Conj. 1: (Solved)
For any non-negative real number $r > 0$, there exists a set, $K$ of which strict Hausdorff Dimension is equal to $r$.

Fields: Measure Theory, Real Analysis
Proposed: 02.24.15
Solved: 03.16.15
The general outline is presented in the post, CONSTRUCTING A SET WITH ARBITRARY STRICT HAUSDORFF DIMENSION

Conj. 2: (Unsolved)
Let $p$ be a prime. Let $r_p$ denote the smallest primitive root of $(\mathbb{Z}/p\mathbb{Z})^\times$. Then, $\displaystyle \liminf_{p \to \infty} r_p$ is bounded, while $\displaystyle \limsup_{p \to \infty} r_p$ diverges.

Fields: Elementary Number Theory
Proposed: 10.28.14

Conj. 3: (Solved)
Let $\mathbb{P}$ be a set of primes. The supremum of $s$ such that $\displaystyle \sum_{p \in \mathbb{P}} \frac{1}{p^x}$ diverges is $x = 1$.

Fields: Analytic Number Theory
Proposed: 02.26.16
Solved: 03.17.16
The general outline is presented in the post, The Half-Plane of Convergence of Zeta Function, and it begot a new conjecture, Conj. 4.

Conj. 4: (Unsolved)
Let us define Prime Zeta Function as $\displaystyle P(s) = \sum_{p \in \mathbb{P}} \frac{1}{p^s}$ where $\mathbb{P}$ is a set of primes. The domain of $P(s)$ is all complex plane except $s = 1$ like Riemann Zeta Function.

Fields: Analytic Number Theory
Proposed: 03.17.16

The post is still in progress.