Apostol 해석적 정수론 CH3. 28~31

이건 또 정수론 공부할 때 (2017.08.09) 열심히 쓰던 글이다. Inline math 가독성 극혐... 나중에 TeX으로 써서 따로 올려야겠다. 일단은 포스팅이 적혀 있는 것까지만.

추가 notation : 

$$ \gamma := \lim_{n \to \infty} \left( \sum_{k=1}^{n}\frac{1}{k} - \text{log} n \right) \text{ (Euler - Mascheroni const.) } $$

$$ \zeta(s) := \begin{cases} \sum_{k=1}^{\infty} \frac{1}{k^s} & {(s>1)} \\ \lim_{x \rightarrow \infty} {\left(\sum_{n \le x} \frac{1}{k^s} - \frac{x^{1-s}}{1-s} \right)} & {(0<s<1)} \end{cases} $$

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D28. Arithmetic mean of a function

$$ \tilde{f}(n) := \frac{1}{n}\sum_{k=1}^{n}f(k) $$

D29. Big-Oh notation, Asymptotic equality, Error Term

$$ f(x) = O(g(x)) \Rightarrow \exists M>0 \text{ s.t. } M\cdot g(x) \ge | f(x) | $$

$$ f(x) \sim g(x) \text{ as } x \rightarrow \infty \Rightarrow \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = 1 $$

$$ \text{When } f(x) \sim g(x) \Rightarrow E_{f,g}(x) := |f(x)-g(x)| $$

T30. Euler's Summation Formula

$$ \text{if } f\in \mathbb{C}^1([y,x]) \ (f' \text{ is continuous} ) \\ \sum_{y<n \le x}f(n) = \int_{y}^{x} f(t)dt + \int_{y}^{x} (t-[t])f'(t)dt + f(x)([x]-x) - f(y)([y]-y) $$

pf) 열심히 부분적분인데 쓰면 길어서 생략. 힌트는 가우스 붙은 적분의 구간을 쪼개는 것.

T31. Harmonic Series.

$$ \sum_{n \le x} \frac{1}{n} = \text{log}x + \gamma + O(\frac{1}{x}) $$

pf)

$$ \text{Lemma) } \\ \gamma = 1-\int_{1}^{\infty}\frac{t-[t]}{t^2}dt$$

pf of Lemma)

$$ \int_{1}^{\infty}\frac{t-[t]}{t^2}dt \\ = \int_{1}^{\infty}\frac{1}{t} - \int_{1}^{\infty}\frac{[t]}{t^2}dt \\ = \lim_{n \rightarrow \infty} \left(\text{log}n - \int_{1}^{n}\frac{[t]}{t^2}dt\right) \\ = \lim_{n \rightarrow \infty} \left(\text{log}n - \sum_{k=1}^{n-1}\int_{k}^{k+1}\frac{k}{t^2}dt \right) \\ = \lim_{n \to \infty} \left(\text{log}n - \sum_{k=2}^{n+1}\frac{1}{k} \right) \\ = 1-\gamma \ \blacksquare $$

$$ \text{by T30(Euler's SF),} \\ \sum_{n \le x} \frac{1}{n} = \sum_{1<n \le x} \frac{1}{n} + 1$$

$$ = 1 + \int_{1}^{x}\frac{1}{t}dt - \int_{1}^{x}\frac{t-[t]}{t^2}dt - \color{blue}{\frac{x-[x]}{x}} $$

$$ = \text{log}x +1 \color{magenta}{-\int_{1}^{x}\frac{t-[t]}{t^2}dt } \color{blue}{+O(\frac{1}{x})} $$

$$ = \text{log}x + 1 \color{magenta}{-\int_{1}^{\infty}\frac{t-[t]}{t^2}dt} \color{red}{+ \int_{x}^{\infty}\frac{t-[t]}{t^2}dt + O(\frac{1}{x})} $$

$$ = \text{log}x + \color{green}{1 - \int_{1}^{\infty}\frac{t-[t]}{t^2}dt} + \color{red}{O(\frac{1}{x})} \\ \left(\because \int_{x}^{\infty}\frac{t-[t]}{t^2}dt < \int_{x}^{\infty}\frac{1}{t^2}dt = \frac{1}{x} \right) \\ =_{\text{Lemma}} \quad \text{log}x + \color{green}{\gamma} + O(\frac{1}{x}) \ \blacksquare $$