RAMANUJAN's SUM

Srinivasa Ramanujan

\[C_{q}(n)\]

\[C_{q}(n)=\sum_{a=1~with~(a,q)=1}^{q}~~e^{2\pi i a \frac{n}{q}}\]


#Ramanujan's sum

def ramanujanC(q, n) :
    a = var('a')
    j = IntegerRing()
    j = sum([exp(2*pi*I*a*n/q) for a in range(q) if gcd(a,q) == 1]).numerical_approx(digits=1).real()
    return j


ramanujanC(12, 12)

[ramanujanC(n, n) for n in [1..10]]
[ramanujanC(n, 1) for n in [1..10]]
[ramanujanC(1, n) for n in [1..10]]

\[C_{n}(n)=\varphi (n)\]

\[C_{q}(1)=\mu (q)\]

A054533 Triangular array giving Ramanujan sum.

1,-1,1,-1,-1,2,0,-2,0,2,-1,-1,-1,-1,4,1,-1,-2,-1,1,2,-1,-1,-1
A054533    OEIS

A054534 Square array giving Ramanujan sum.

1,1,-1,1,1,-1,1,-1,-1,0,1,1,2,-2,-1,1,-1,-1,0,-1,1,1,1,-1,2,-1
A054534    OEIS

A054535 Square array giving Ramanujan sum.

1,-1,1,-1,1,1,0,-1,-1,1,-1,-2,2,1,1,1,-1,0,-1,-1,1,-1,-1,-1,2
A054535    OEIS