Apostol 해석적 정수론 CH2. (20~27)

1편 : http://drugstoreoftamref.tistory.com/15

2편 : http://drugstoreoftamref.tistory.com/16

I have no time!

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D20. Generalized Dirichlet Convolution

$$ \text{for } x \in \mathbb{R}, \\ (a \circ F)(x) := \sum_{n \le x} a(n)F(\frac{x}{n}) $$

\(\text{cf) if } \forall x\notin \mathbb{Z} \quad F(x)=0 \ \Rightarrow \ \circ \equiv \ast \)

T21. $$ a \circ (b \circ F) = (a \ast b) \circ F $$

pf)

\( a \circ (b \circ F) (x) \)

\( = \sum_{n \le x} a(n) \cdot \left( \sum_{m \le x/n} b(m)F(x/nm) \right) \)

\( = \sum_{nm \le x} a(n)b(m)F(x/nm) \)

\( = \sum_{k \le x} \left( \sum_{d|k} a(d)b(k/d) \right) \cdot F(x/k) \)

\( = (a \ast b) \circ F (x) \ \blacksquare \)

T22. Generalized Inversion Formula

$$ \text{if } a(1)\neq 0 \text{ (there exists a Dirichlet inversion of a)} \\ G=a \circ F \Leftrightarrow F = a^{-1} \circ G $$

pf)

\( a^{-1} \circ G = a^{-1} \circ (a \circ F) = (a^{-1} \ast a) \circ F = I \circ F = F \ \blacksquare \)

D23. Bell Series 

$$ f_p (x) := \sum_{k=0}^{\infty} f(p^n)x^n $$

MF의 성질을 분석하는 데 유용하다고 함.

ex) \(I_p (x) = 1, \ \mu_p (x) = 1-x, \ N_p (x) = \frac{1}{1-px} \)

cf) \(f \in \mathbb{CM} \Rightarrow f_p (x) = \frac{1}{1-f(p)x} \)

T24. Uniqueness Theorem

$$ f = g  \ \Leftrightarrow \ \forall p \quad f_p(x) = g_p(x) $$

pf) 소인수분해를 이용하면 자명.

T25. $$ (f \ast g)_p (x) = f_p(x) \cdot g_p(x) $$

pf)

\( (f \ast g)(p^k) = \sum_{d|n} f(d)g(p^k/d) = \sum_{i=0}^{k} f(p^i)g(p^{k-i}) \ \blacksquare \)

D26. Derivative of Arithmetic Functions

$$ f' := f\cdot \log $$

cf) 미분연산자의 일반적인 조건이 성립

\( f' + g' = (f+g)' \)

\( (f \ast g)' = (f' \ast g) + (f \ast g') \)

\( (f^{-1})' = -f' \ast (f \ast f)^{-1} \approx -\frac{f'}{f^2}\)

cf 2)

\( u' = \log , \ I' = 0 , \Lambda \ast u = \log = u' \)

T27. Selberg's Identity

\( \text{cf : An important lemma to prove PNT} \)

$$ \Lambda(n)\log (n) + \sum_{d|n}\Lambda(d)\Lambda(\frac{n}{d}) = \sum_{d|n} \mu (d) \log^2 (\frac{n}{d}) $$

pf)

\( \text{ The identity is equivalent to } \)

\( \Lambda ' + \Lambda \ast \Lambda = \mu \ast u'' \)

양변에 \(u\)를 합성하면 등식은

\( \Lambda' \ast u + \Lambda \ast (\Lambda \ast u) = u'' \)

\( \Leftrightarrow_{\text{D26 cf2}} \ \Lambda' \ast u + \Lambda \ast u' = u'' \)

\( \Leftrightarrow_{\text{D26 cf1}} \ (\Lambda \ast u)' = u'' \ \blacksquare \)

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Exercise는 시간 될 때!