Summation and product over $k$ with $k$ prime to $n$ sought

Question

I just come to a standstill with the following two formulas.

If $$E_n=\lbrace k\mid 1\le k\le n\ \&\ (k,n)=1\rbrace$$ then I hope for a closed formula $f(n)$ for those

Would be welcome if the Euler totient $\phi$ function would appear in $f(n)$.

For $(2),$ see http://mathworld.wolfram.com/CyclotomicPolynomial.html — lab bhattacharjee, Aug 13 '14 at 17:02
The first should be available here: http://math.stackexchange.com/questions/569210/sum-of-all-coprimes-of-a-number — lab bhattacharjee, Aug 13 '14 at 17:02
The cyclotomic polynomials deal with $e^{\frac{2\pi ik}{n}}$ but I am looking for $\cos(\frac{k\pi}{n})$ without the factor of $2$. — Wolfgang Tintemann, Aug 13 '14 at 17:17

score 0 · Answer 1 · answered Aug 13 '14 at 18:08

0

For (2) I made a calculation for $n\ge 3$ and get

$$\prod_{E_n}\cos\left(\frac{k\pi}{n}\right)=\frac{\Phi_n(-1)}{2^{\phi(n)}}$$

This formula may be totally wrong at this moment.

I have not the CAS Mathematica available and also not the programming skills to check it by other CAS freely available.

answered Aug 13 '14 at 18:08

Maybe I missed a factor of $(-1)^{\frac{\phi(n)}{2}}$ on the right side as the formula is not correct and I dont find the error in my calculation. To calculate $\phi(n)$ by logarithm it would be sufficient to take the absolute value on the left side. – Wolfgang Tintemann Aug 14 '14 at 13:29

1 Answers1