WolframAlpha says that $\sum^n_{k-1}\left\lfloor \frac{n}{k} \right\rfloor \phi(k) \not = \sum^n_{k=1}k$ which is not true

Question

I submitted the following query to WolframAlpha:

Is sum Floor[n/k]Phi[k], k=1 to n equal to sum k, k to n

and the result (link) was:

$\sum^n_{k-1}\left\lfloor \frac{n}{k} \right\rfloor \phi(k)$ is not always equal to $\sum^n_{k=1}k$

The interesting thing is that it is not true according to the solutions presented in:

Euler phi Function and Floor Function.
Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$.

What happened? Is it a bug, or there are some edge cases that WolframAlpha is taking under account?

Mathematica (and therefore WolframAlpha I presume) is not able to simplify the sum down to $n(n+1)/2$ so thats probably the reason it returns false. Returning "I don't know" would probably have been a better answer. — Winther, Sep 02 '16 at 15:44
@Winther This is why I am asking. WolframAlpha should not say that they are not always equal. — Mateusz Piotrowski, Sep 02 '16 at 15:45
@N.S. Perfect, I guess I just have to notify the WolframAlpha dev team now. — Mateusz Piotrowski, Sep 02 '16 at 15:47
I think that this mistake was pointed to them couple years back, still not fixed :https://www.wolframalpha.com/input/?i=derivative+floor+x — N. S., Sep 02 '16 at 15:48
@N.S. Interesting. Even the newest Mathematica (version 11) gives strange values for the derivative of the floor function at non-integer values. For example $\text{Floor}'(1.1) = -4.29$ apposed to $0$. — Winther, Sep 02 '16 at 16:08

score 1 · Accepted Answer · answered Sep 05 '16 at 12:42

I've submitted this bug to the WolframAlpha team.

This is their response:

We appreciate your feedback regarding Wolfram|Alpha. The issue you reported has been passed along to our development team for review. Thank you for helping us improve Wolfram|Alpha.

WolframAlpha says that $\sum^n_{k-1}\left\lfloor \frac{n}{k} \right\rfloor \phi(k) \not = \sum^n_{k=1}k$ which is not true

1 Answers1