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A limit with Euler's totient function

Real Analysis
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Tolaso J Kos
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A limit with Euler's totient function


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

$$\ell = \lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{k=1}^{n} \sin \left (\frac{\pi k}{n} \right) \varphi(k)$$
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Re: A limit with Euler's totient function


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

We are quoting a theorem by Omran Kouba:

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 128 times
Hence the limit is $\frac{6}{\pi^3}$.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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