Toto sem pridávam z podobných dôvodov, prečo som sem pridal post o "solutions manuals". Neplánujeme síce odpisovať riešenia z inokade, ale ak si nebudeme s niečím vedieť poradiť, tak sa možno nakoniec pozrieme na to, ako to vyriešil niekto iný.
Na rozdiel od toho druhého postu (kde by sme chceli mať linky na zbierky obsahujúce veľa riešených príkladov, sem môžeme dávať, keď niekde nájdeme linky na riešenia jednotlivých cvičení. (Možno nejaké nájdete náhodou, keď budete na nejakej stránke s matematickým obsahom z úplne iného dôvodu. A tiež sa dá niekedy vo voľnom čase hrať s hľadaním matiky - hľadanie matematických výrazov na webe nie je úplne jednoduché, môže byť zaujímavé si to trochu vyskúšať. Aj na tomto fóre som spomenul nejaké príklady.)
Zdá sa, že dosť veľa vecí sa dá nájsť takto: Apostol "number theory" site:math.stackexchange.com
Prípadne môžete vyskúšať nejaké iné matematické fóra, ktoré poznáte, napríklad:
* Apostol "number theory" site:site:mathoverflow.net
* Apostol "number theory" site:mymathforum.com OR site:physicsforums.com
Apostol - riešenia jednotlivých cvičení online
Moderator: Martin Sleziak
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Martin Sleziak
- Posts: 5686
- Joined: Mon Jan 02, 2012 5:25 pm
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Martin Sleziak
- Posts: 5686
- Joined: Mon Jan 02, 2012 5:25 pm
Re: Apostol - riešenia jednotlivých cvičení online
Chapter 2
Exercise 1: 1. Find all integers such that:
(a) $\varphi(n)=\frac n2$
(b) $\varphi(n)=2\varphi(n)$
(c) $\varphi(n)=12$
1a:
http://math.stackexchange.com/questions ... 2-iff-n-2k
http://math.stackexchange.com/questions ... -then-n-2j
http://math.stackexchange.com/questions ... t-phin-n-2
1b:
http://math.stackexchange.com/questions ... i2n-phin1c
1c:
http://math.stackexchange.com/questions ... at-phin-12
3. Prove that
$$\frac n{\varphi(n)}=\sum_{d\mid n} \frac{\mu^2(d)}{\varphi(d)}.$$
http://math.stackexchange.com/questions ... -mu2d-phid
5. Define $\nu(1)=0$, and for $n>1$ let $\nu(n)$ be the number of distinct prime factors of $n$. Let $f=\mu*\nu$ and prove that $f(n)$ is either $0$ or $1$.
https://math.stackexchange.com/q/3096612
14. Let $f(x)$ be defined for all rational $x$ in $0\le x \le 1$ and let
$$F(n)=\sum_{k=1}^n f\left(\frac kn\right), \qquad F^*(n)=\sum_{\substack{k=1\\(k,n)=1}}^n f\left(\frac kn\right).$$
(a) Prove that $F^*=\mu*F$, the Dirichlet product of $\mu$ and $F$.
(b) Use (a) or some other means to prove that $\mu(n)$ is the sum of the primitive $n$th roots of unity:
$$\mu(n)=\sum_{\substack{k=1\\(k,n)=1}}^n e^{2\pi ik/n}.$$
14b:
http://math.stackexchange.com/questions ... unction-mu
Exercise 1: 1. Find all integers such that:
(a) $\varphi(n)=\frac n2$
(b) $\varphi(n)=2\varphi(n)$
(c) $\varphi(n)=12$
1a:
http://math.stackexchange.com/questions ... 2-iff-n-2k
http://math.stackexchange.com/questions ... -then-n-2j
http://math.stackexchange.com/questions ... t-phin-n-2
1b:
http://math.stackexchange.com/questions ... i2n-phin1c
1c:
http://math.stackexchange.com/questions ... at-phin-12
3. Prove that
$$\frac n{\varphi(n)}=\sum_{d\mid n} \frac{\mu^2(d)}{\varphi(d)}.$$
http://math.stackexchange.com/questions ... -mu2d-phid
5. Define $\nu(1)=0$, and for $n>1$ let $\nu(n)$ be the number of distinct prime factors of $n$. Let $f=\mu*\nu$ and prove that $f(n)$ is either $0$ or $1$.
https://math.stackexchange.com/q/3096612
14. Let $f(x)$ be defined for all rational $x$ in $0\le x \le 1$ and let
$$F(n)=\sum_{k=1}^n f\left(\frac kn\right), \qquad F^*(n)=\sum_{\substack{k=1\\(k,n)=1}}^n f\left(\frac kn\right).$$
(a) Prove that $F^*=\mu*F$, the Dirichlet product of $\mu$ and $F$.
(b) Use (a) or some other means to prove that $\mu(n)$ is the sum of the primitive $n$th roots of unity:
$$\mu(n)=\sum_{\substack{k=1\\(k,n)=1}}^n e^{2\pi ik/n}.$$
14b:
http://math.stackexchange.com/questions ... unction-mu
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Martin Sleziak
- Posts: 5686
- Joined: Mon Jan 02, 2012 5:25 pm
Re: Apostol - riešenia jednotlivých cvičení online
Chapter 3:$\newcommand{\dcc}[1]{\lfloor#1\rfloor}$
2. If $x\ge 2$ prove that:
$$\sum_{n\le x} \frac{d(n)}n = \frac12 \log x + 2C\log x+O(1),$$
where $C$ is Euler's constant.
https://math.stackexchange.com/question ... -summation
3. If $x\ge 2$ and $\alpha>0$, $\alpha\ne1$, prove that $$\sum_{n\le x} \frac{d(n)}{n^\alpha}=\frac{x^{1-\alpha}\ln x}{1-\alpha}+\zeta(\alpha)^2+O(x^{1-\alpha}).$$
https://math.stackexchange.com/question ... x-fracdnna
4. If $x\ge 2$ prove that:$\newcommand{\ldcc}[1]{{\left\lfloor #1 \right\rfloor}}$
(a) $\sum_{n\leq x} \mu(n)\ldcc{\frac xn}^2=\frac{x^2}{\zeta(2)}+O(x\log x)$
(b) $\sum_{n\le x} \mu(n)\ldcc{\frac xn} = \frac{x}{\zeta(2)}+O(\log x)$
4a:
https://math.stackexchange.com/question ... ummatory-f https://math.stackexchange.com/question ... problem-4a
https://math.stackexchange.com/question ... r-2-fracx2
6. If $x\ge 2$ prove that
$$\sum_{n\le x} \frac{\varphi(n)}{n^2} = \frac1{\zeta(2)}\log x+\frac{C}{\zeta(2)}-A+O\left(\frac{\log x}x\right),$$
where $C$ is Euler's constant and
$$A=\sum_{n=1}^\infty \frac{\mu(n)\log n}{n^2}.$$
https://math.stackexchange.com/question ... -x-zeta2-f
https://math.stackexchange.com/question ... -varphinn2
https://math.stackexchange.com/question ... ac-log-nn2
https://math.stackexchange.com/question ... -functions
9a: https://math.stackexchange.com/question ... -pi26-frac
$$\frac{\sigma(n)}n<\frac{n}{\varphi(n)}<\frac{\pi^2}6\frac{\sigma(n)}n.$$
9b: https://math.stackexchange.com/question ... cn-phin-ox
$$\sum\limits_{n\leq x}\frac{n}{\varphi(n)}=O(x)$$
10: https://math.stackexchange.com/question ... thcalo-log
$$\sum_{n\le x}\frac1{\varphi(n)}=O(\log x).$$
11b: https://math.stackexchange.com/question ... n-mud-sigm
$$\varphi_1(n)= \sum_{d^2\mid n}\mu(d)\sigma\left(\frac n{d^2}\right)$$
12: https://math.stackexchange.com/question ... me-to-some
$$\sum_{\substack{n\le x\\(n,k)=1}} \frac1{n^s}$$
16b: https://math.stackexchange.com/question ... y-geq-xyxy
https://math.stackexchange.com/question ... n-identity
https://math.stackexchange.com/question ... oor-lfloor
$$\lfloor2x\rfloor+\lfloor2y\rfloor \ge \lfloor x\rfloor + \lfloor y\rfloor + \lfloor x+y\rfloor$$
17: Prove that $\dcc x + \dcc{x+\frac12}=\dcc{2x}$ and, more generally,
$$\sum_{k=0}^{n-1} \ldcc{x+\frac kn} = \dcc{nx}.$$
https://math.stackexchange.com/question ... d-nx-sum-k
https://math.stackexchange.com/question ... -0n-1-left
20: $\dcc{\sqrt n+\sqrt{n+1}}=\dcc{\sqrt{4n+2}}$
https://math.stackexchange.com/question ... oor-sqrt4n
https://math.stackexchange.com/question ... prove-that
21. Find all positive integers such that $\dcc{\sqrt{n}}$ divides $n$.
https://math.stackexchange.com/question ... rfloor-mid
2. If $x\ge 2$ prove that:
$$\sum_{n\le x} \frac{d(n)}n = \frac12 \log x + 2C\log x+O(1),$$
where $C$ is Euler's constant.
https://math.stackexchange.com/question ... -summation
3. If $x\ge 2$ and $\alpha>0$, $\alpha\ne1$, prove that $$\sum_{n\le x} \frac{d(n)}{n^\alpha}=\frac{x^{1-\alpha}\ln x}{1-\alpha}+\zeta(\alpha)^2+O(x^{1-\alpha}).$$
https://math.stackexchange.com/question ... x-fracdnna
4. If $x\ge 2$ prove that:$\newcommand{\ldcc}[1]{{\left\lfloor #1 \right\rfloor}}$
(a) $\sum_{n\leq x} \mu(n)\ldcc{\frac xn}^2=\frac{x^2}{\zeta(2)}+O(x\log x)$
(b) $\sum_{n\le x} \mu(n)\ldcc{\frac xn} = \frac{x}{\zeta(2)}+O(\log x)$
4a:
https://math.stackexchange.com/question ... ummatory-f https://math.stackexchange.com/question ... problem-4a
https://math.stackexchange.com/question ... r-2-fracx2
6. If $x\ge 2$ prove that
$$\sum_{n\le x} \frac{\varphi(n)}{n^2} = \frac1{\zeta(2)}\log x+\frac{C}{\zeta(2)}-A+O\left(\frac{\log x}x\right),$$
where $C$ is Euler's constant and
$$A=\sum_{n=1}^\infty \frac{\mu(n)\log n}{n^2}.$$
https://math.stackexchange.com/question ... -x-zeta2-f
https://math.stackexchange.com/question ... -varphinn2
https://math.stackexchange.com/question ... ac-log-nn2
https://math.stackexchange.com/question ... -functions
9a: https://math.stackexchange.com/question ... -pi26-frac
$$\frac{\sigma(n)}n<\frac{n}{\varphi(n)}<\frac{\pi^2}6\frac{\sigma(n)}n.$$
9b: https://math.stackexchange.com/question ... cn-phin-ox
$$\sum\limits_{n\leq x}\frac{n}{\varphi(n)}=O(x)$$
10: https://math.stackexchange.com/question ... thcalo-log
$$\sum_{n\le x}\frac1{\varphi(n)}=O(\log x).$$
11b: https://math.stackexchange.com/question ... n-mud-sigm
$$\varphi_1(n)= \sum_{d^2\mid n}\mu(d)\sigma\left(\frac n{d^2}\right)$$
12: https://math.stackexchange.com/question ... me-to-some
$$\sum_{\substack{n\le x\\(n,k)=1}} \frac1{n^s}$$
16b: https://math.stackexchange.com/question ... y-geq-xyxy
https://math.stackexchange.com/question ... n-identity
https://math.stackexchange.com/question ... oor-lfloor
$$\lfloor2x\rfloor+\lfloor2y\rfloor \ge \lfloor x\rfloor + \lfloor y\rfloor + \lfloor x+y\rfloor$$
17: Prove that $\dcc x + \dcc{x+\frac12}=\dcc{2x}$ and, more generally,
$$\sum_{k=0}^{n-1} \ldcc{x+\frac kn} = \dcc{nx}.$$
https://math.stackexchange.com/question ... d-nx-sum-k
https://math.stackexchange.com/question ... -0n-1-left
20: $\dcc{\sqrt n+\sqrt{n+1}}=\dcc{\sqrt{4n+2}}$
https://math.stackexchange.com/question ... oor-sqrt4n
https://math.stackexchange.com/question ... prove-that
21. Find all positive integers such that $\dcc{\sqrt{n}}$ divides $n$.
https://math.stackexchange.com/question ... rfloor-mid
-
Martin Sleziak
- Posts: 5686
- Joined: Mon Jan 02, 2012 5:25 pm
Re: Apostol - riešenia jednotlivých cvičení online
Chapter 4
10. Let $s_n$ denote the $n$th partial sum of the series
$$\sum_{r=1}^\infty \frac1{r(r+1)}.$$
Prove that for every integer $k>1$ there exist integers $m$ and $n$ such that $s_m-s_n=1/k$.
http://math.stackexchange.com/questions ... ven-series
10. Let $s_n$ denote the $n$th partial sum of the series
$$\sum_{r=1}^\infty \frac1{r(r+1)}.$$
Prove that for every integer $k>1$ there exist integers $m$ and $n$ such that $s_m-s_n=1/k$.
http://math.stackexchange.com/questions ... ven-series