Ramanujan's sum
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- This article is not about Ramanujan summation.
In mathematics, Ramanujan's sum, named for Srinivasa Ramanujan and usually denoted cq(n), is defined to be
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:
[edit] Series involving Ramanujan's sum
Ramanujan evaluated infinite series of the form
for several sequences (aq).[1] In particular, for s any real number greater than or equal to 1, he found that the Dirichlet series
where σ is the divisor function and ζ the Riemann zeta function. In the cases s = 1 and s = 2 this yields
and
respectively.
Other identities obtained by Ramanujan are
and
where r2(n) denotes the number of representations of n as x2 + y2 in integers x and y.
[edit] References
- ^ Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, G. H. Hardy, Cambridge University Press, 1940
