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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|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|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|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|Additive number theory]]); for example, the representation of a natural number as an even number of squares. S. Ramanujan [[#References|[1]]] obtained many formulas involving Ramanujan sums.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. Ramanujan,   "On certain trigonometrical sums and their applications in the theory of numbers"  ''Trans. Cambridge Philos. Soc.'' , '''22'''  (1918)  pp. 259–276  ((Also: Collected papers, Chelsea, reprint, 1962.))</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,   "Note on Ramanujan's trigonometrical function $c_q(n)$ and certain series of arithmetical functions"  ''Proc. Cambridge Philos. Soc.'' , '''20'''  (1920–1921)  pp. 263–271</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy (ed.)  et al. (ed.) , ''Collected papers of S. Ramanujan'' , Chelsea, reprint  (1962)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  B. Volkmann,   "Verallgemeinerung eines Satzes von Maxfield"  ''J. Reine Angew. Math.'' , '''271'''  (1974)  pp. 203–213</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  E.C. Titchmarsh,   "The theory of the Riemann zeta-function" , Clarendon Press  (1951) {{ZBL|0042.07901}}</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Levin,   "The life and work of the Indian mathematician Ramanujan"  ''Istoriko-Mat. Issled.'' , '''13'''  (1960)  pp. 335–378  (In Russian)</TD></TR>
</table>