Welcome to mathimatikoi.org forum; Enjoy your visit here.

A limit with Euler's totient function

Real Analysis
Post Reply
User avatar
Tolaso J Kos
Administration team
Administration team
Articles: 2
Posts: 854
Joined: Sat Nov 07, 2015 6:12 pm
Location: Larisa
Contact:

A limit with Euler's totient function

#1

Post by Tolaso J Kos » Wed Apr 19, 2017 11:34 am

Here is something I created.

Let $\varphi$ denote Euler’s totient function. Evaluate the limit

$$\ell = \lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{k=1}^{n} \sin \left (\frac{\pi k}{n} \right) \varphi(k)$$
Imagination is much more important than knowledge.
User avatar
Riemann
Articles: 0
Posts: 169
Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

Re: A limit with Euler's totient function

#2

Post by Riemann » Sat May 26, 2018 1:18 pm

We are quoting a theorem by Omran Kouba:
Theorem:

Let $\alpha$ be a positive real number and let $\{a_n\}_{n \in \mathbb{N}}$ be a sequence of positive real numbers such that

$$\lim_{n \rightarrow +\infty} \frac{1}{n^\alpha} \sum_{k=1}^{n} a_k = \ell$$

For every continuous function $f$ on the interval $[0, 1]$ it holds that

$$\lim_{n \rightarrow +\infty} \frac{1}{n^\alpha} \sum_{k=1}^{n} f \left ( \frac{k}{n} \right ) a_k = \ell \int_{0}^{1} \alpha x^{\alpha-1} f(x) \, {\rm d}x$$
Proof: The theorem can be found at the attachment following:
MR_1_2010 Riemann Sums(kouba).pdf
(97.32 KiB) Downloaded 33 times
Hence the limit is $\frac{6}{\pi^3}$.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
Post Reply