Ramanujan's Sum - Ramanujan Expansions

Ramanujan Expansions

If f(n) is an arithmetic function (i.e. a complex-valued function of the integers or natural numbers), then a convergent infinite series of the form

or of the form

(where the a_k are complex numbers),

is called a Ramanujan expansion of f(n). .

Ramanujan found expansions of some of the well-known functions of number theory. All of these results are proved in an "elementary" manner (i.e. only using formal manipulations of series and the simplest results about convergence).

The expansion of the zero function depends on a result from the analytic theory of prime numbers, namely that the series converges to 0, and the results for r(n) and r′(n) depend on theorems in an earlier paper.

All the formulas in this section are from Ramanujan's 1918 paper.