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A limit with Euler's totient function

Posted: Wed Apr 19, 2017 11:34 am
by Tolaso J Kos
Here is something I created.

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

$$\ell = \lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{k=1}^{n} \sin \left (\frac{\pi k}{n} \right) \varphi(k)$$

Re: A limit with Euler's totient function

Posted: Sat May 26, 2018 1:18 pm
by Riemann
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 127 times
Hence the limit is $\frac{6}{\pi^3}$.