Math Problems Vault

Problem VI.3.1

(a)(b)(c)(d)

Let $Q>1, p, q \in \mathbb {R}$ . Prove that as $n \rightarrow +\infty$ the following relations hold

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(a)

$\sum_{1 \leqslant k \leqslant n} k^{p} \ln^{q} k \sim \frac{n^{p+1}}{p+1} \ln^{q} n \quad (p>-1)$ ;

(b)

$\sum_{k \geqslant n} \frac{\ln^{q} k}{k^{p}} \sim \frac{\ln^{q} n}{(p-1) n^{p-1}} \quad (p>1)$ ;

(c)

$\sum_{1 \leqslant k \leqslant n} k^{p} Q^{k} \sim \frac{Q}{Q-1} n^{p} Q^{n}$ ;

(d)

$\sum_{k \geqslant n} \frac{k^{p}}{Q^{k}} \sim \frac{Q}{Q-1} \frac{n^{p}}{Q^{n}}$ .

Problem VI.3.2

Let $\alpha >0$ and $x_{n}=\sum_{1 \leqslant k \leqslant n^{\alpha }} \frac{1}{k}\left(1-\frac{1}{n}\right)^{k}$ . Prove that $x_{n} \sim \min (1, \alpha ) \ln n$ .

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Problem VI.3.3

(a)(b)

Let $a_{n}>0, b_{n}=o\left(a_{n}\right), c_{n} \sim a_{n}$ . Prove that

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(a)

if $\sum a_{n}=+\infty$ , then

\sum _{1 \leqslant k \leqslant n} b_{k}=o\left(\sum _{1 \leqslant k \leqslant n} a_{k}\right), \quad \sum _{1 \leqslant k \leqslant n} c_{k} \sim \sum _{1 \leqslant k \leqslant n} a_{k} ;

(b)

if $\sum a_{n}<+\infty$ , then

\sum _{k \geqslant n} b_{k}=o\left(\sum _{k \geqslant n} a_{k}\right), \quad \sum _{k \geqslant n} c_{k} \sim \sum _{k \geqslant n} a_{k} .

Remark.

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Recall that $\sum a_{n}$ denotes the series $\sum_{n \geqslant 1} a_{n}$ .

Problem VI.3.4

(a)(b)(c)(d)

Let $a_{n} \downarrow 0, p \geqslant 0, \alpha \in \mathbb {R}$ . Prove that

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(a)

if $\sum_{1 \leqslant k \leqslant n} k^{\alpha } a_{k}=O\left(n^{p}\right)$ , then $a_{n}=O\left(n^{p-\alpha -1}\right)$ ;

(b)

if $\sum_{1 \leqslant k \leqslant n} k^{\alpha } a_{k}=o\left(n^{p}\right)$ , then $a_{n}=o\left(n^{p-\alpha -1}\right)$ ;

(c)

if $\quad \sum_{k \geqslant n} k^{\alpha } a_{k}=O\left(n^{-p}\right)$ , then $a_{n}=O\left(n^{-p-\alpha -1}\right)$ ;

(d)

if $\quad \sum_{k \geqslant n} k^{\alpha } a_{k}=o\left(n^{-p}\right)$ , then $a_{n}=o\left(n^{-p-\alpha -1}\right)$ .

Problem VI.3.5

(a)(b)(c)

Let $p>0, \alpha \in \mathbb {R}$ and $\sum_{1 \leqslant k \leqslant n} k^{\alpha } x_{k} \sim n^{p}$ . Prove that

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(a)

if $p>\alpha$ , then $S_{n}=x_{1}+\ldots +x_{n} \sim \frac{p}{p-\alpha } n^{p-\alpha }$ ;

(b)

if $p=\alpha$ , then $S_{n}=x_{1}+\ldots +x_{n} \sim p \ln n$ ;

(c)

if $p<\alpha$ , then the series $\sum x_{n}$ converges and $\sigma_{n}=\sum_{k \geqslant n} x_{k} \sim \frac{p}{\alpha -p} n^{p-\alpha }$ .

Problem VI.3.6

(a)(b)(c)

Let $p>0, \quad \alpha \in \mathbb {R} \quad$ and $\quad$ the series $\sum k^{\alpha } x_{k} \quad$ converge, with $\sum_{k \geqslant n} k^{\alpha } x_{k} \sim n^{-p}$ . Prove that

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(a)

if $p<-\alpha$ , then $S_{n}=x_{1}+\ldots +x_{n} \sim \frac{p}{\left|p+\alpha \right|} n^{\left|p+\alpha \right|}$ ;

(b)

if $p=-\alpha$ , then $S_{n}=x_{1}+\ldots +x_{n} \sim p \ln n$ ;

(c)

if $p>-\alpha$ , then the series $\sum x_{n}$ converges and $\sigma_{n}=\sum_{k \geqslant n} x_{k} \sim \frac{p}{p+\alpha } n^{-(p+\alpha )}$ .

Problem VI.3.7

(a)(b)

Let $a_{n}>0$ and $\sqrt[n]{a_{n}} \uparrow +\infty$ . Prove that

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(a)

$\sum_{1 \leqslant k \leqslant n} k^{p} a_{n} \sim n^{p} a_{n}$ ;

(b)

$\sum_{k \geqslant n} \frac{k^{p}}{a_{k}} \sim \frac{n^{p}}{a_{n}}$ .

Problem VI.3.8

(a)(b)

Let $a_{n} \downarrow 0, S_{n}=a_{1}+\ldots +a_{n} \sim \frac{n^{1-p}}{1-p}$ , where $0<p<1$ .

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(a)

Prove that $a_{n} \sim n^{-p}$ .

(b)

Can one claim that $a_{n} \sim \frac{1}{n}$ , if $S_{n} \sim \ln n$ ?

Problem VI.3.9

Let $a_{n} \downarrow 0$ and $\sum a_{n}<+\infty$ , with $\sum_{k \geqslant n} a_{k} \sim n^{-p}$ , where $p>0$ . Prove that $a_{n} \sim p n^{-1-p}$ .

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Problem VI.3.10

(a)(b)

Let $a_{n} \downarrow 0$ . Prove that

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(a)

$\varlimsup_{\varepsilon \cdot \rightarrow +0} \frac{1}{\ln (1 / \varepsilon )} \sum a_{n}^{1+\varepsilon }=\varlimsup_{N \rightarrow \infty } \frac{1}{\ln \ln \left(1 / a_{N}\right)} \sum_{1 \leqslant n \leqslant N} a_{n}$ ;

(b)

$\overline{\lim }_{\varepsilon \rightarrow +0} \varepsilon \sum a_{n}^{1+\varepsilon } \leqslant \overline{\lim }_{N \rightarrow \infty } \frac{1}{\ln \left(1 / a_{N}\right)} \sum_{1 \leqslant n \leqslant N} a_{n} \leqslant e \overline{\lim }_{\varepsilon \rightarrow +0} \varepsilon \sum a_{n}^{1+\varepsilon }$ .

Can the coefficient $e$ in inequality b) be decreased?

Problem VI.3.11

Let $\alpha_{n}>0, \sum \alpha_{n}<+\infty$ and $f(t)=t \sum \frac{\alpha_{n}}{1+t^{2} \alpha_{n}}$ for $t \geqslant 0$ . Prove that the behavior of the function $f$ as $t \rightarrow +\infty$ is closely related to the behavior of the sequence $\left\{ n \sqrt{\alpha_{n}}\right\}$ :

\frac{\pi }{2} \lim n \sqrt{\alpha _{n}} \leqslant \lim _{t \rightarrow +\infty } f(t) \leqslant \varlimsup _{t \rightarrow +\infty } f(t) \leqslant \frac{\pi }{2} \overline{\lim } n \sqrt{\alpha _{n}}

Moreover,

\sup _{t \geqslant 0} f(t) \leqslant \frac{\pi }{2} \sup n \sqrt{\alpha _{n}}

and the coefficient $\frac{\pi }{2}$ cannot be decreased.

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Problem VI.3.12

Let $f:[0,+\infty ) \rightarrow \mathbb {R}$ be a nonnegative nonincreasing function, $h>0$ . Prove that

\lim _{h \rightarrow +0} h \sum f(k h)=\int _{0}^{+\infty } f(t) d t

Is this true if the function $f$ is continuous but not monotonic?

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Problem VI.3.13

(a)(b)(c)

Determine the asymptotics of the following sums as $A \rightarrow +\infty$ :

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(a)

$\sum_{k \geqslant 1} \frac{1}{k^{2}+A^{2}}$ ;

(b)

$\sum_{k \geqslant 1} e^{-k^{2} / A^{2}}$ ;

(c)

$\sum_{k \geqslant 1} \frac{k^{p-1}}{k^{q}+A^{q}}(0<p<q)$ .

Problem VI.3.14

(a)(b)

Let the function $f$ be nonnegative and decreasing on $[1,+\infty )$ . Prove that

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(a)

if $\int_{1}^{+\infty } f(t) d t=+\infty$ , then

\sum _{1 \leqslant k \leqslant n} f(k)=\int _{1}^{n+1} f(t) d t+C+o(1), \quad 0 \leqslant C \leqslant f(1)

in particular $\sum_{1 \leqslant k \leqslant n} f(k) \sim \int_{1}^{n} f(t) d t$ ;

(b)

if $\int_{1}^{+\infty } f(t) d t<+\infty$ , then

\sum _{k \geqslant n} f(k)=\int _{n}^{+\infty } f(t) d t+O(f(n))

in particular, if $f(x)=o\left(\int_{x}^{+\infty } f(t) d t\right)$ as $x \rightarrow +\infty$ , then

\sum _{k \geqslant n} f(k) \sim \int _{n}^{+\infty } f(t) d t

Problem VI.3.15

If the function $f$ is monotonic on $\left[m, n\right](m, n \in \mathbb {Z})$ , then

\left|\sum _{m \leqslant k \leqslant n} f(k)-\int _{m}^{n} f(t) d t\right| \leqslant \max (\left|f(n)\right|,\left|f(m)\right|)

In particular, if the function $f$ is monotonic on $[1, \infty ), \int_{1}^{\infty } f(t) d t=+\infty$ and

f(x)=o\left(\int _{1}^{x} f(t) d t\right) \text{ as } x \rightarrow +\infty , \text{ then } \sum _{1 \leqslant k \leqslant n} f(k) \sim \int _{1}^{n} f(t) d t .

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Problem VI.3.16

(a)(b)(c)

Let $M, N \in \mathbb {Z}, M \leqslant N$ , let the function $f$ (in general, complex-valued) be integrable on the interval $\left[M-\frac{1}{2}, N+\frac{1}{2}\right]$ , and let

\Delta =\sum _{M \leqslant k \leqslant N} f(k)-\int _{M-1 / 2}^{N+1 / 2} f(t) d t

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(a)

Prove that $\left|\Delta \right| \leqslant \frac{1}{2} \underset {M-\frac{1}{2}}{\stackrel{N+\frac{1}{2}}{\mathrm{~ V}}}(f)$ . In particular, if $f$ is monotonic, then

\left|\Delta \right| \leqslant \frac{1}{2}\left|f\left(N+\frac{1}{2}\right)-f\left(M-\frac{1}{2}\right)\right| .

(b)

If $f \in C^{1}\left(\left[M-\frac{1}{2}, N+\frac{1}{2}\right]\right)$ , then

\Delta =\int _{M-1 / 2}^{N+1 / 2}\left(t-\left[t\right]-\frac{1}{2}\right) f^{\prime }(t) d t

(c)

If $f \in C^{2}\left(\left[M-\frac{1}{2}, N+\frac{1}{2}\right]\right)$ , then

\Delta =-\frac{1}{2} \int _{M-1 / 2}^{N+1 / 2}\left(t-\left[t\right]-\frac{1}{2}\right)^{2} f^{\prime \prime }(t) d t

and consequently, $\left|\Delta \right| \leqslant \frac{1}{8} \int_{M-1 / 2}^{N+1 / 2}\left|f^{\prime \prime }(t)\right| d t$ . In particular, if $f$ is convex or concave, then $\left|\Delta \right| \leqslant \frac{1}{8}\left|f^{\prime }\left(N+\frac{1}{2}\right)-f^{\prime }\left(M-\frac{1}{2}\right)\right|$ .

Problem VI.3.17

Let $f(p)=\sum_{n \geqslant 1} \frac{(-1)^{n+1}}{n^{p}}$ for $p>0$ . Prove that

f(p)=\frac{1}{2}+O(p) .

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Problem VI.3.18

Prove that as $p \rightarrow +0$

\sum _{n \geqslant 1} \frac{1}{n^{p+1}}=\frac{1}{p}+\gamma +o(1) \quad (\gamma -\text{ Euler's constant}) .

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Problem VI.3.19

Prove that

\begin{aligned} \prod _{1 \leqslant k \leqslant n}\left(1+\frac{1}{\sqrt{k}}\right) & \sim A \frac{\exp (2 \sqrt{n})}{\sqrt{n}}, \\ \prod _{1 \leqslant k \leqslant n}\left(1+\frac{1}{\sqrt[3]{k}}\right) & \sim B \sqrt[3]{n} \exp \left(\frac{3}{2}\left(n^{2 / 3}-n^{1 / 3}\right)\right) \end{aligned}

for some positive numbers $A$ and $B$ .

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Problem VI.3.20

Let $\alpha \in \mathbb {R} \backslash \mathbb {N}$ . Prove that

\binom {n-\alpha }{n}=\frac{(n-\alpha )(n-\alpha -1) \ldots (1-\alpha )}{n!} \sim \frac{C_{\alpha }}{n^{\alpha }} \quad \left(C_{\alpha } \neq 0\right) .

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Problem VI.3.21

(a)(b)(c)

Prove that

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(a)

$\sqrt[n^{2}]{1 \cdot 2^{2} \cdot 3^{3} \ldots n^{n}} \sim \frac{\sqrt{n}}{\sqrt[4]{e}}$ ;

(b)

$\sqrt[n^{3}]{1!\cdot (2!)^{2} \cdot (3!)^{3} \ldots (n!)^{n}} \sim \frac{\sqrt[3]{n}}{e^{4 / 9}}$ ;

(c)

$\sqrt[n]{1 \cdot 2^{2} \cdot 3^{3} \ldots n^{n}} \sim \left(\frac{n}{\sqrt{e}}\right)^{n / 2} \sqrt{n}$ .

Problem VI.3.22

(a)(b)(c)(d)

Let $\alpha >0$ . Prove that as $n \rightarrow +\infty (n \in \mathbb {N})$

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(a)

$\sum_{1 \leqslant k \leqslant n} k^{\alpha k / n} \sim \frac{n^{1+\alpha }}{\alpha \ln n}$ ;

(b)

$\sum_{1 \leqslant k \leqslant n} k^{-\alpha k / n} \sim \frac{n}{\alpha \ln n}$ ;

(c)

$\sum_{1 \leqslant k \leqslant n}(k!)^{-\alpha / n} \sim \frac{n}{\alpha \ln n}$ ;

(d)

$\sum_{1 \leqslant k \leqslant n} k^{\alpha n} \sim \frac{n^{\alpha n}}{1-e^{-\alpha }}$ .

Problem VI.3.23

Prove that $\sum_{1 \leqslant k \leqslant n} \frac{n^{k}}{k!} \sim \frac{1}{2} e^{n}$ .

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Problem VI.3.24

(a)(b)

Prove that

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(a)

$\sum_{1 \leqslant k \leqslant n} \sqrt{C_{n}^{k}} \sim \sqrt[4]{2 \pi n} 2^{n / 2}$ ;

(b)

$\sum_{1 \leqslant k \leqslant n} C_{n}^{k} \sqrt{\left|2 k-n\right|} \sim \frac{\Gamma \left(\frac{3}{4}\right)}{\sqrt{\pi }} \sqrt[4]{2 n} 2^{n}$ .

Problem VI.3.25

Let $N \in \mathbb {N}$ and $x>0$ . Prove that

\sum _{n \geqslant N} e^{-x \sqrt{n}}=\left(\frac{2}{x} \sqrt{N-\frac{1}{2}}+\frac{2}{x^{2}}+O\left(\frac{x}{\sqrt{N}}\right)\right) e^{-x \sqrt{N-1 / 2}}

where the constant in the $O$ -estimate does not depend on $x$ and $N$ .

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Problem VI.3.26

(a)(b)(c)

Let the sequence of numbers $\left\{ x_{n}\right\}$ be such that $\frac{x_{n}}{\sqrt{n}} \rightarrow C$ , $0 \leqslant C \leqslant +\infty$ . Find the asymptotics of the sum

\alpha _{N}=\sum _{n \geqslant N} e^{-x_{N} \sqrt{n}}

in the following cases:

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(a)

$C=0$ ;

(b)

$C=+\infty$ ;

(c)

$0<C<+\infty$ .

Problem VI.3.27

Let $p>0$ and $S(x)=\sum_{k=0}^{\left[x\right]}(x-k)^{p}$ . Prove that for some $C>0$ , for all $x>0$ the inequality holds

\left|S(x)-\frac{1}{p+1} x^{p+1}\right| \leqslant C x^{p} .

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Problem VI.3.28

Let $\psi \in C^{1}([a,+\infty )), \psi , \psi^{\prime }>0$ and $\psi =O\left(\psi^{\prime }\right)$ on $[a,+\infty )$ . Prove that

\int _{a}^{+\infty } e^{-\varepsilon \psi (t)} d t=\psi ^{-1}\left(\frac{1}{\varepsilon }\right)+O(1) \quad \text{ as } \quad \varepsilon \rightarrow +0

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Problem VI.3.29

Find the asymptotics as $t \rightarrow 1-0$ of the sums of the series

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(a)

$\sum \frac{1}{n} t^{\ln^{p} n}(p>0)$ ;

(b)

$\sum t^{\ln^{p} n} \quad (p>1)$ ;

(c)

$\sum t^{n^{p}} \quad (p>0)$ ;

(d)

$\sum t^{a^{n}} \quad (a>1)$ ;

(e)

$\sum t^{(a n)^{n}}(a>0)$ ;

(f)

$\sum t^{n!}$ .

Problem VI.3.30

(a)(b)

Find the asymptotics as $t \rightarrow 1-0$ of the sums of the series

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(a)

$\sum n^{p} t^{n}(p>-1)$ ;

(b)

$\sum_{n \geqslant 2}(\ln n)^{p} t^{n}(p \in \mathbb {R})$ .

Problem VI.3.31

Find the asymptotics as $t \rightarrow 1-0$ of the sums of the series

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(a)

$\sum \frac{t^{n}}{1+t^{n}}$ ;

(b)

$\sum \frac{n t^{n}}{1+t^{n}}$ ;

(c)

$\quad \sum \frac{t^{n}}{\left(1+t^{n}\right)^{2}}$ ;

(d)

$\quad \sum \frac{n t^{n}}{\left(1+t^{n}\right)^{2}}$ ;

(e)

$\sum \frac{t^{n}}{1-t^{n}}$ ;

(f)

$\sum \frac{n t^{n}}{1-t^{n}}$ ;

(g)

$\sum \frac{t^{n}}{\left(1-t^{n}\right)^{2}}$ ;

(h)

$\sum \frac{n t^{n}}{\left(1-t^{n}\right)^{2}}$ ;

(i)

$\quad \sum \frac{(-1)^{n} t^{n}}{1-t^{n}}$ ;

(j)

$\quad \sum \frac{(-1)^{n} n t^{n}}{1-t^{n}}$ ;

(k)

$\quad \sum \frac{(-1)^{n} t^{n}}{\left(1-t^{n}\right)^{2}}$ ;

(l)

$\quad \sum \frac{(-1)^{n} n t^{n}}{\left(1-t^{n}\right)^{2}}$ .

Problem VI.3.32

Find the asymptotics as $t \rightarrow 1-0$ of the sums of the series

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(a)

$\sum \frac{t^{n}}{1+t^{2 n}}$ ;

(b)

$\sum \frac{t^{n}}{\left(1+t^{2 n}\right)^{2}}$ ;

(c)

$\quad \sum \frac{n t^{n}}{1-t^{2 n}}$ ;

(d)

$\quad \sum \frac{t^{n}}{1-t^{2 n}}$ ;

(e)

Problem VI.3.33

Let $\tau (k)$ be the number of divisors of the natural number $k$ . Prove that

\sum _{1 \leqslant k \leqslant n} \tau (k)=n(\ln n+2 \gamma -1)+O(\sqrt{n}) .

Here $\gamma$ is Euler's constant (see problem II.2.8).

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Problem VI.3.34

(a)(b)

Find the asymptotics as $t \rightarrow 1-0$ of the sums of the series

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(a)

$T(t)=\sum \tau (n) t^{n}$ ,

(b)

$S(t)=\sum \sigma (n) t^{n}$ , $(\tau (n)$ is the number of divisors, $\sigma (n)$ is the sum of divisors of the number $n$ ).

Problem VI.3.35

(a)(b)

Let a smooth function $\varphi$ be convex or concave on $[1,+\infty )$ , $\lim_{x \rightarrow +\infty } \varphi (x)=+\infty$ . Prove that

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(a)

if $\ln \varphi (x)=o(x)$ as $x \rightarrow +\infty$ , then

\sum \varphi (n) t^{n} \sim \frac{1}{1-t} \sum \varphi ^{\prime }(n) t^{n} \quad (t \rightarrow 1-0) ;

(b)

if $\varphi (x)=O(x)$ as $x \rightarrow +\infty$ , then

\sum \varphi (n) t^{n}=\frac{1}{1-t}\left(\sum \varphi ^{\prime }(n) t^{n}+C+o(1)\right) \quad (t \rightarrow 1-0),

where $C=\lim \left(\varphi (n)-\left(\varphi^{\prime }(1)+\ldots +\varphi^{\prime }(n)\right)\right)$ . In particular, taking $\varphi (x)=\ln x$ , we obtain a refinement of the result of problem 3.30 b) with $p=1$ :

\sum t^{n} \ln n=\left|\ln (1-t)\right|-\gamma +o(1)

Problem VI.3.36

(a)(b)

Find the asymptotics as $A \rightarrow +\infty$ of the sums ( $C>0$ )

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(a)

$\sum e^{-C n}\left(1-\frac{1}{n^{2}}\right)^{A}$ ;

(b)

$\sum e^{-C n}\left(1-\frac{1}{n}\right)^{A}$ .

Problem VI.3.37

(a)(b)

Let $1<b<a$ . Prove that as $s \rightarrow +\infty$

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(a)

$\sum_{n \geqslant 1} \frac{b^{n}}{a^{n}+a^{s}} \sim \left(\frac{b}{a}\right)^{s} \varphi (s)$ ;

(b)

$\sum_{n \geqslant 1} \frac{1}{a^{n} \operatorname {sh}\left(b^{s-n}\right)} \sim \frac{\psi (s)}{a^{s}}$ ,

where $\varphi , \psi -1$ -periodic positive smooth functions:

\varphi (s)=\sum _{k \in \mathbb {Z}} \frac{b^{k-s}}{1+a^{k-s}}, \quad \psi (s)=\sum _{k \in \mathbb {Z}} \frac{a^{k-s}}{\operatorname {sh}\left(b^{k-s}\right)} .

What is the mean value of the function $\psi$ when $a=b^{2}$ ?

Problem VI.3.38

Find the asymptotics of the sum

S_{m}=\sum _{n \geqslant 0} \frac{1}{2^{n}}\left(1-\frac{1}{2^{n}}\right)^{2^{m}}

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Problem VI.3.39

Prove that

\max _{x_{1}^{2}+\ldots +x_{n}^{2}=1} \sum _{k=1}^{n}\left(x_{1}+\frac{x_{2}}{\sqrt{2}}+\ldots +\frac{x_{k}}{\sqrt{k}}\right)^{2} \underset {n \rightarrow +\infty }{\sim } n \ln n .

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Problem VI.3.40

(a)(b)(c)

Let $s>1,\left\{ p_{k}\right\}$ be the sequence of all prime numbers, numbered in increasing order. For an odd index $2 n+1$ set $\varphi (2 n+1)=(-1)^{n}$ .

Arguing in the same way as when solving problems I.2.11 b) and I.2.12 b), prove that

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(a)

$\prod_{k \geqslant 2}\left(1-\frac{\varphi \left(p_{k}\right)}{p_{k}^{s}}\right)^{-1}=\sum_{n \geqslant 0} \frac{(-1)^{n}}{(2 n+1)^{s}}$ ;

(b)

$\sum_{k \geqslant 1} \frac{\varphi \left(p_{k}\right)}{p_{k}^{s}}=O(1)$ .

(c)

Using the result of problem I.2.12b), derive from this the asymptotic relations

\sum _{p_{k}=1 \bmod 4} \frac{1}{p_{k}^{s}} \underset {s \rightarrow 1}{\sim } \frac{1}{2} \ln \frac{1}{s-1}, \quad \sum _{p_{k}=-1 \bmod 4} \frac{1}{p_{k}^{s}} \underset {s \rightarrow 1}{\sim } \frac{1}{2} \ln \frac{1}{s-1} .

From part c) it follows that each of the arithmetic progressions $\left\{ 4 m \pm 1\right\}$ contains infinitely many primes. This is a special case of the celebrated Dirichlet theorem, according to which any arithmetic progression $\left\{ a m+b\right\}$ with coprime natural numbers $a$ and $b$ contains infinitely many primes (see [GL], [Ka], [Tr]). Interesting information related to this theorem can be found in $[\mathrm{Kn}]$ . It is also apparent that the progressions $\left\{ 4 m \pm 1\right\}$ contain "approximately equal numbers" of primes. This result (as well as problem I.2.15) can be supplemented by the following asymptotic relations.

Problem VI.3.41

Prove that

\sum _{\substack {p_{k}=1 \bmod 4 \\ p_{k} \leqslant N}} \frac{1}{p_{k}} \underset {N \rightarrow \infty }{\sim } \frac{\ln \ln N}{2}, \quad \sum _{\substack {p_{k}=-1 \bmod 4 \\ p_{k} \leqslant N}}^{\frac{1}{p_{k}}} \underset {N \rightarrow \infty }{\sim } \frac{\ln \ln N}{2} .

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