Ramanujan's sum

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In mathematics, Ramanujan's sum, named for Srinivasa Ramanujan and usually denoted cq(n), is defined to be

c_q(n)=\sum_{a=1\atop (a,q)=1}^qe\left(\frac{an}{q}\right),

where n and q are positive integers, (a,q) denotes the greatest common divisor of a and q, and e(x) is the exponential function exp(2πix).

It is easily shown that Ramanujan's sum is multiplicative, i.e.

cq(n)cr(n)=cqr(n)

whenever (q,r) = 1.

Another property is that cq(n) equals its complex conjugate, hence is real.

Writing d for the greatest common divisor of q and n, and denoting the Möbius function and Euler's totient function by μ and φ respectively, one has the following identity:

c_q(n)=\mu(q/d)\frac{\phi(q)}{\phi(q/d)}.

[edit] Series involving Ramanujan's sum

Ramanujan evaluated infinite series of the form

\sum_{q=1}^\infty a_qc_q(n)

for several sequences (aq).[1] In particular, for s any real number greater than or equal to 1, he found that the Dirichlet series

\sum_{q=1}^\infty\frac{c_q(n)}{q^s}=\frac{\sigma_{1-s}(n)}{\zeta(s)},

where σ is the divisor function and ζ the Riemann zeta function. In the cases s = 1 and s = 2 this yields

\sum_{q=1}^\infty\frac{c_q(n)}{q}=0

and

\sum_{q=1}^\infty\frac{c_q(n)}{q^2}=\frac{6}{\pi^2}\frac{\sigma_1(n)}{n}

respectively.

Other identities obtained by Ramanujan are

\sum_{q=1}^\infty\frac{c_q(n)}{q}\log(q)=-\sigma_0(n)

and

\sum_{q=1}^\infty(-1)^{q-1}\frac{c_{2q-1}(n)}{2q-1}=r_2(n),

where r2(n) denotes the number of representations of n as x2 + y2 in integers x and y.


[edit] References

  1. ^ Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, G. H. Hardy, Cambridge University Press, 1940
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