Does $f(x)=\frac1x\int_0^x\frac{1}{\sin t+\sin (\pi t)+2}dt$ have a supremum?

Asked Sep 24 '23 at 05:23

Active Sep 24 '23 at 06:35

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(This question was inspired by What is the mean value of $|\sin x +\sin (\pi x)|$?.)

My thoughts:

$\frac{1}{\sin t+\sin (\pi t)+2}$ can be arbitrarily large, but that doesn't tell me whether its average value from $t=0$ to $t=x$, for all $x\in\mathbb{R}$, has a supremum.

Wolfram does not evaluate $\int \frac{1}{\sin t+\sin (\pi t)+2}dt$.

Here is the graph of $y=\frac1x\int_0^x\frac{1}{\sin t+\sin (\pi t)+2}dt$ from $x=-30$ to $x=30$.

edited Sep 24 '23 at 06:35

asked Sep 24 '23 at 05:23

Dan

22,158

I believe this is an open problem. If $f(x)$ does have a supremum, it should imply the irrational measure of $\pi$, let's call it $\mu$, is $2$. As of this moment, no one know what the exact value of $\mu$ is, we only know $2 \le \mu \le 7.1032..$ (ref) – achille hui Sep 24 '23 at 07:07
Why did you remove your last question about $\sup(\frac{\tan n}n)$? – Michael Morad Sep 28 '23 at 17:08
@MichaelMorad It was answered by Concerning the sequence $\Big(\dfrac {\tan n}{n}\Big) $. – Dan Sep 28 '23 at 22:54

Does $f(x)=\frac1x\int_0^x\frac{1}{\sin t+\sin (\pi t)+2}dt$ have a supremum?

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