$\sum_{j=1}^{n}\left(\left(\frac{jk}{n}\right)\right)=\frac{n-1}{2}$ if $(n,k)=1$.
How could we show this identity?
$\sum_{j=1}^{n}\left(\left(\frac{jk}{n}\right)\right)=\frac{n-1}{2}$ if $(n,k)=1$.
How could we show this identity?
Iām assuming that the upper limit of the summation should be $n$; with $k$ there the result is false.
For $j=1,\dots,n$ write $jk=a_jn+r_j$, where $a_j$ and $r_j$ are integers and $0\le r_j