# Ramanujan sums

Jump to: navigation, search

Trigonometric sums depending on two integer parameters $k$ and $n$:

$$c _ {k} ( n) = \sum _ { h } \mathop{\rm exp} \left ( \frac{2 \pi n h i }{k} \right ) = \ \sum _ { h } \cos \frac{2 \pi n h }{k} ,$$

when $h$ runs over all non-negative integers less than $k$ and relatively prime to $k$. The basic properties of Ramanujan sums are multiplicity with respect to the index $k$,

$$c _ {k k ^ \prime } ( n) = c _ {k} ( n) c _ {k ^ \prime } ( n) \ \ \textrm{ if } ( k , k ^ \prime ) = 1 ,$$

and also the representation in terms of the Möbius function $\mu$:

$$c _ {k} ( n) = \ \sum _ {d \mid ( k , n ) } \mu \left ( \frac{k}{d} \right ) d .$$

Ramanujan sums are finite if $k$ or $n$ is finite. In particular, $c _ {k} ( 1) = 1$.

Many multiplicative functions on the natural numbers (cf. Multiplicative arithmetic function) can be expanded as series of Ramanujan sums, and, conversely, the basic properties of Ramanujan sums enable one to sum series of the form

$$\sum _ { n= } 1 ^ \infty \frac{c _ {k} ( q n ) }{n ^ {s} } f ( n) ,\ \ \sum _ { k= } 1 ^ \infty \frac{c _ {k} ( q n ) }{k ^ {s} } f ( k) ,$$

where $f$ is a multiplicative function and $q$ is an integer. In particular,

$$\sum _ { k= } 1 ^ \infty \frac{c _ {k} ( n) }{n ^ {s} } = \ \frac{\sigma _ {1-} s ( n) }{\zeta ( s) } ,$$

where $\zeta$ is the Riemann zeta-function and $\sigma _ {a}$ is the sum of the $a$- th powers of the divisors of $n$. Such sums are closely connected with special series for certain additive problems in number theory (cf. Additive number theory); for example, the representation of a natural number as an even number of squares. S. Ramanujan  obtained many formulas involving Ramanujan sums.

How to Cite This Entry:
Ramanujan sums. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ramanujan_sums&oldid=48417
This article was adapted from an original article by K.Yu. Bulota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article