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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 Eulerâ€™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:

Hence the limit is $\frac{6}{\pi^3}$.